Life on Voyager
During the survey of a proto-nebula in the Delta Quadrant, the away team's shuttlecraft became ensnared in the gravimetric shear of a plasma surge, disabling the craft's propulsion systems. After receiving the shuttle's distress signal, Voyager carried out an emergency beam-out to retrieve the crew comprising Tom Paris, B'Elanna Torres, Seven of Nine, and The Doctor.
Because of the effects of the proto-nebula, transporter operator Ensign Mulchaey initially experienced trouble establishing a good pattern lock however despite having to manually separate the individual patterns, he was able to re-materialize the team safely.
Unfortunately, The Doctor's mobile emitter, actually a 29th century piece of technology, was heavily damaged during transport and some of the nanoprobes were unknowingly incorporated into its circuitry. The emitter was taken to Voyager's science lab where it was subsequently assimilated by Seven's nanoprobes, and in turn began assimilating the lab.
When Mulchaey was sent to check up on the status of the emitter, he was assaulted with extraction tubules from the emitter. However, contrary to standard Borg assimilation procedures, Mulchaey himself was not assimilated into a drone. Instead, the tubules extracted a sample of DNA from the officer. Within the assimilated lab the Borg technology constructed a maturation chamber, where a Borg fetus grew out of the DNA sample. Within hours the fetus had matured into a fully grown Borg drone, a rate of development far out-pacing standard Borg maturation cycles, with the mobile emitter running his neocortex.
As a combined product of 29th century and Borg technology, the new drone possessed several unique features and abilities. These included internal transporter nodes, multi-dimensional adaptability, reactive body armor composed of the same poly-deutonic alloy as used in the mobile emitter, and a multi-spatial personal force field. In essence, it was a 29th century drone, immensely more powerful than 24th century Borg.
Despite the risks, Captain Kathryn Janeway decided to keep this particular drone alive in an attempt to have more help against the Borg, or at least learn more about them. She ordered Seven of Nine to take charge of the drone when it emerged from the maturation chamber and requested its designation and assignment. Unsure what to call the drone, Seven refused to assign it a designation, stating that it was "irrelevant", but Neelix wanted to give it a name, assigning it the designation "One", and Seven set about instructing it on various topics through a neural link. As this almost resulted in all of her knowledge taken by the drone, she used Borg data nodes. In short order One was able to assimilate 47 billion teraquads of information. As he learned more, the drone predictably began to inquire about the Borg, and at one point even asked about joining the Collective.
Under Captain Janeway's directions, Seven attempted to dissuade One from seeking further information about the Collective and warned him that joining the Collective would mean losing his individuality. However, One inadvertently alerted a Borg sphere with a second subspace transmitter shortly after his first was deactivated, and it was soon detected approaching Voyager's position. Faced with a confrontation, Janeway ordered Seven to show One the true nature of the Borg. Seven examined the data with One who, despite his desire to experience the hive mind, felt threatened by the Borg's destructive nature and their desire to destroy Voyager and assimilate its crew of individuals.
As the Borg sphere approached and began to tractor in Voyager, One made a decision to help the crew resist the attack. His advanced 29th century technology gave One an advantage and he modified the shields within moments to disengage the tractor. A modification of the phasers was not as successful, and One was forced to board the sphere directly knowing he was superior and they would fail to capture him. Once aboard the sphere, One was successful in warding off attacking drones and interfacing himself with the Borg to control the ship's navigation. From there, he propelled the sphere into the same stellar phenomenon that led to his birth, which destroyed the vessel. One was fortunate to escape, using his personal shield around himself as he floated in space.
However, his biological components were badly injured. The Doctor tried to treat One's injuries, but One refused to allow this, activating his personal force field; he knew the Borg would hunt Voyager to assimilate him for his technology, and One was a grave risk to the ship as long as he was alive. Seven displayed a closeness with One, much like a mother; when One died on the operating table, Seven was greatly saddened by his passing. Afterwards, the mobile emitter was retrieved for The Doctor. (VOY: "Drone")
"Terminate interface! You must comply... you are hurting me."
"...I will comply"
- - Seven of Nine and One
"Seven of Nine tells me designation is irrelevant."
"I disagree. You should choose a name for yourself. Something that defines who you are. After all, there's only one of you."
- - One and Neelix
"On the contrary. Our primary mission is to explore new forms of life. You may have been unexpected but, given time, I am sure you'll make a fine addition to the crew. After all, you've got my mobile emitter driving your neocortex, so you're bound to make a dazzling impression. That's called a joke."
"Joke. A verbal comment or gesture designed to provoke laughter."
"I see you've got your mother's sense of humor."
- - The Doctor and One
"To date I've assimilated forty seven billion teraquads of information on a vast variety of subjects, including particle physics, comparative humanoid anatomy, warp field theory, and the culinary delights of the Delta Quadrant. "
- - One
"You must comply."
"I will not."
"You must comply. Please... you are hurting me."
"You will adapt."
- - Seven of Nine and One, after One refused The Doctor's treatment to save his life
One was played by recurring Star Trek guest actor J. Paul Boehmer, who later recalled that One was his favorite role on par with Mestral in "Carbon Creek". "The most fun I've had working on Star Trek has been playing One and Mestral both." he said in a StarTrek.com interview. "One because the Borg are such an interesting character-villain and also because he was a different Borg. He was not typical average mean guy Borg villain. He had to be matured and natured by the crew and really kind of claim who he was throughout the episode. So it left it open to a lot of possibility. It wasn't just this set way of being a Borg. It lent itself to some more humanity. And that was really exciting because Star Trek historically has expanded the envelope of so many things. And to see them expand the envelope of their own villain and what the possibilities there with this mean mean villain and have it be open to being something else was really exciting to be a part of that." 
The creation of One's physical appearance involved multiple processes. "I had to go through the whole prosthetic routine [....] It was a four hour make-up session," Boehmer remembered. "I had to get a body cast made [....] I was in the body cast for two hours. It was also a really long process of getting in and out of the suit I was wearing [....] The artists had four days to design and put together this costume." Boehmer didn't find these processes arduous, instead referring to the prosthetic routine as "totally cool." He continued, "The guys were great [....] I was warned that [the costume] was going to be very claustrophobic and unbearable, but I loved it all. If I had to do it every week, I'm sure that it would lose its charm, but for the week that I did it, it was terrific fun [....] What [the artists who designed and built the costume] came up with is nothing short of amazing. I was totally blown away by that." (Star Trek Monthly issue 46, p. 71)
The costume also comprised a special neon eyepiece instead of blinking LEDs, a reduced number of tubes to reflect the advanced Borg technology, a different lighting system for his suit and a special Borg-type appliance that covered the right ear. A small two-inch neon light was used for the eye that had to be constructed large enough for the neon fixture and to accommodate the wires running from the appliance to the battery pack on Boehmer's back. (Star Trek: Aliens and Artifacts, p. 163)
David A. McIntee wrote, "One is simply wonderful. [...] If only they could have kept him for a little longer. [...] The real revelation is J Paul Boehmer as the drone – he's stunning. [...] The expressions that cross his face when he assimilates the data node are almost worth the admission price alone. Though another 'innocent' regular would be a bad move, it's a shame they couldn't hang on to him for an arc of three or four episodes." (Delta Quadrant, p.256)
In "Brief Candle", a short story in the Distant Shores anthology, it is revealed that One's remains were used to create the multi-spatial probe, explaining why the crew always went to great lengths to recover it as it was literally irreplaceable.