Satellite galaxies of the Milky Way are among the most promising targets for dark matter searches in gamma rays. We present a search for dark matter consisting of weakly interacting massive particles, applying a joint likelihood analysis to 10 satellite galaxies with 24 months of data of the Fermi Large Area Telescope. No dark matter signal is detected. Including the uncertainty in the dark matter distribution, robust upper limits are placed on dark matter annihilation cross sections. The 95% confidence level upper limits range from about 1e-26 cm^3 s^-1 at 5 GeV to about 5e-23 cm^3 s^-1 at 1 TeV, depending on the dark matter annihilation final state. For the first time, using gamma rays, we are able to rule out models with the most generic cross section (~3e-26 cm^3 s^-1 for a purely s-wave cross section), without assuming additional boost factors. Show Partial Abstract