The first space shuttle to dock with the Russian space station Mir was Atlantis. It achieved that milestone during STS-71, which launched from Kennedy Space Center on June 27, 1995, and docked with Mir on June 29.
STS-71 was the first mission in the Shuttle–Mir program to complete an actual shuttle-station docking. Atlantis delivered cosmonauts Anatoly Solovyev and Nikolai Budarin to Mir, while bringing astronaut Norman Thagard and cosmonauts Vladimir Dezhurov and Gennady Strekalov back to Earth. The linked spacecraft formed the largest object ever assembled in orbit at that time.
A common mix-up is Discovery: it flew STS-63, the earlier “near-Mir” mission, and became the first shuttle to rendezvous with Mir without docking. Atlantis later flew seven consecutive Shuttle–Mir missions. STS-71 also marked the first shuttle crew exchange with an orbiting station and the first U.S.–Russian space-station docking since Apollo–Soyuz in 1975.