The lift off took place at 17:02 local time (21:02 GMT) on Saturday and marked the third shuttle launch this year and the 35th for Discovery. The shuttle is expected to dock with the ISS two days after launch, NASA said.
The 14-day mission will see the Discovery shuttle deliver its heaviest payload to the world's sole orbiter, the Japanese Pressurized Module (JPM) which is the second unit of the massive Kibo laboratory complex.
The first unit, an experimental logistics module, was delivered to the ISS in March.
Three spacewalks are also planned during the shuttle crew's nine-day stay on the ISS installing the Kibo's JPM along with a remote manipulator system (RMS) comprising two robotic arms.
The STS-124 shuttle crew is commanded by Mark Kelly, 44, with Ken Ham serving as pilot. Mission specialists include Karen Nyberg, Ronald Garan, Michael Fossum and Japanese Aerospace Exploration Agency (JAXA) astronaut Akihiko Hoshide.
The seventh crew member Greg Chamitoff will replace Garrett Reisman, who has been on the ISS for three months.
