The International Space Station was a space station and one of Earth's first internationally combined efforts in exploring space. The station was jointly serviced by American space shuttle orbiters and Russian Soyuz and Progress spacecraft.
In 2000, astronaut Lieutenant McMillan was scheduled to be the co-pilot on a joint mission between NASA and the Europeans for a four month tour on the International Space Station that began in 2003. (VOY: "11:59")
Beginning with the year 2370, Benjamin Sisko had a model in his office aboard Deep Space 9 of the ISS with the space shuttle Enterprise docked. The model was housed in a glass casing in the beginning but was later displayed without the casing in his office next to Ops. (Star Trek: Deep Space Nine)
The model of the International Space Station seen in Star Trek: Deep Space Nine was given to the series by Majel Barrett as a gift. The DS9 art department added a model of the Enterprise (OV-101) and displayed it docked to the ISS. (Star Trek Encyclopedia (4th ed., vol. 1, p. 372)) 
The names Gidzenko, Krikalyov, and Shepherd were names seen on a mission insignia in the 602 Club. They were the first inhabitants of the space station on Expedition 1. Their mission patch, seen in the 602 Club, showed the space station on a blue background and the names of the Russian astronauts were written in their native Cyrillic alphabet.
E. Michael Fincke, who appeared, along with fellow NASA astronaut Terry Virts in ENT: "These Are the Voyages...", has spent a year, total, aboard the ISS, from April to October 2004 (speaking with Scott Bakula while in space) and again from October 2008 to April 2009 - Virts himself piloted the shuttle Endeavour on a mission to the station in February 2010. He later returned to the station in November 2014 as Flight Engineer of Expedition 42/43.
According to the reference work Ships of the Line (p. 8), in the 2150s, the ISS has remained in Low Earth Orbit and hasn't been destroyed by a controlled re-entry. (In the real world, the ISS is planned for controlled re-entry 2030, a similar fate to MIR)
From the Encyclopedia (4th ed., vol. 1, p. 372), "The International Space Station (ISS) was featured in the main title sequence of Star Trek: Enterprise, beginning with the first episode, "Broken Bow". The decision to add the ISS to the Enterprise titles was a last-minute decision on the part of the producers. Because there was so little time to obtain accurate reference information for the construction of a digital model for visual effects, NASA engineer (and Star Trek fan) Brian J. Young, who at the time was a technician working for the United Space Alliance at NASA's Kennedy Space Center in Florida, went back to work on a Friday night and faxed us the plans for the station!"