The Changeling Julian Bashir was a Changeling of the Founders in the 24th century. In the 2370s, this Changeling was one of a handful of agents tasked with destabilizing relationships between the larger organizations of the Alpha Quadrant.
In 2373, Julian Bashir was captured by the Dominion on Meezan IV and sent to Internment Camp 371. This Founder took his place with the mission of helping Dominion forces travel to Cardassian space and hatching a plan to cripple Federation, Romulan and Klingon forces. It spent a month keeping up appearances before carrying it out, including playing darts and racquetball with Miles O'Brien.
In the meantime, the Dominion had been negotiating incorporating the Cardassian Union, and had to move a fleet into the Alpha Quadrant. After the crew of Deep Space 9 planned to collapse the Bajoran wormhole with a phase-conjugate graviton beam perfected by the Trill Science Ministry, the Bashir-Changeling sabotaged the station's emitter array. This instead stabilized the wormhole and allowed the fleet through.
Later, when Dukat promised to retake DS9 with the new forces, the Changeling attempted to destroy the Bajoran sun, thereby wiping out the combined Federation, Klingon and Romulan fleets gathered to do battle with the Dominion. It stole the runabout the USS Yukon and killed its crewmembers in order to modify it with an explosive device composing of trilithium, tekasite, and protomatter and fly it into the sun. It was, however, intercepted and killed by the USS Defiant before he could carry it out. (DS9: "In Purgatory's Shadow", "By Inferno's Light")
Bashir had previously been impersonated by a Changeling who portrayed Krajensky in "The Adversary". In that episode, however, the real Bashir was missing for a matter of hours before his replacement was revealed to be a spy.
Based on the fact that the real Bashir was not wearing the newer uniforms, which had been introduced in "Rapture", we can deduce that he had been replaced by a Changeling since at least that episode. This means the Changeling performed surgery on Sisko in "Rapture" and tried to help save the Changeling infant in "The Begotten".
On the other hand, it is also possible that it was acceptable to wear either uniform style during that transitional period, as was seen in the transitional period Star Trek Generations, indicating that he may have been captured after the aforementioned events. Furthermore, while it has been argued that is unlikely since all Starfleet officers stationed on DS9 since "Rapture" have been wearing the newer uniforms, the real Bashir clearly gave his whereabouts as being on Meezan IV (not DS9) at the time he was abducted.
In response to the often proposed question, Ronald D. Moore commented "It would've been before "Rapture." (AOL chat, 1997) Moore has also noted that although the writing staff didn't think the revelation would damage anything about previous episodes, René Echevarria did have reservations about the changeling delivering Kira's baby. (AOL chat, 1997)
Star Trek tie-in author, Christopher L. Bennett, proposed the following about the Changeling:
Indeed, the timeline doesn't really work out there. The uniform change was in "Rapture", and Bashir was supposedly abducted 37 days before the end of "By Inferno's Light". But there's no way that "Rapture," "The Darkness and the Light", "The Begotten", "For the Uniform", "In Purgatory's Shadow", and "By Inferno's Light" could've all taken place within less than 37 days. According to dialogue, "Darkness" is at least three weeks before "The Begotten," which in turn covers nearly 2 weeks. So that's at least 33 days right there, and the events of "For the Uniform" and the 2-parter cover about a week each, plus however much time might have elapsed between them. (Even if we assumed 26-hour Bajoran days, 37 of those is only 40 Earth days, so that doesn't help.) So Bashir must have been abducted after the uniform change; there's simply no other possibility. The fact that he's in the old uniform is a paradox.
On the other hand, it's stated in dialogue that Kirayoshi was born "less than a month" before "Purgatory." Since the 2-parter takes about a week, that pretty much requires that Bashir was abducted in between "The Darkness and the Light" and "The Begotten". That part can't be finessed, since the dialogue is explicit. On the one hand, Bashir had to be abducted after the uniform change, which creates a plot hole; but on the other hand, Kirayoshi had to be delivered by the changeling impostor, which creates a second plot hole. It's a total mess -- they just didn't think through the timeline carefully when they put in the date references.
But as for this part 'Because if Changeling Bashir helped to deliver the baby in "The Begotten", then he also did the ultra-delicate brain surgery on Sisko in "Rapture".' Since "Rapture" was significantly more than 37 days before the end of "Inferno," that means it was the real Bashir who did that.