Dwarf spheroidal galaxies are known to be excellent targets for the detection of annihilating dark matter. We present new limits on the annihilation cross section of weakly interacting massive particles based on the joint analysis of seven Milky Way dwarfs using a frequentist Neyman construction and Pass 7 data from the Fermi Gamma-Ray Space Telescope. We exclude generic weakly interacting massive particle candidates annihilating into bb with a mass less than 40 GeV that reproduce the observed relic abundance. To within 95% systematic errors on the dark matter distribution within the dwarfs, the mass lower limit can be as low as 19 GeV or as high as 240 GeV. For annihilation into τ+ τ-, these limits become 19, 13, and 80 GeV, respectively.