We have performed a search for muon neutrinos from dark matter annihilation in the center of the Sun with the 79-string configuration of the IceCube neutrino telescope. For the first time, the DeepCore subarray is included in the analysis, lowering the energy threshold and extending the search to the austral summer. The 317 days of data collected between June 2010 and May 2011 are consistent with the expected background from atmospheric muons and neutrinos. Upper limits are set on the dark matter annihilation rate, with conversions to limits on spin-dependent and spin-independent scattering cross sections of weakly interacting massive particles (WIMPs) on protons, for WIMP masses in the range 20-5000 GeV/c2. These are the most stringent spin-dependent WIMP-proton cross section limits to date above 35 GeV/c2 for most WIMP models.