We study in detail the ability of the nominal configuration of the IceCube neutrino telescope (with 80 strings) to probe the parameter space of the Constrained MSSM (CMSSM) favoured by current collider and cosmological data. Adopting conservative assumptions about the galactic halo model and the expected experiment performance, we find that IceCube has a probability between 2% and 12% of achieving a 5sigma detection of dark matter annihilation in the Sun, depending on the choice of priors for the scalar and gaugino masses and on the astrophysical assumptions. We identify the most important annihilation channels in the CMSSM parameter space favoured by current constraints, and we demonstrate that assuming that the signal is dominated by a single annihilation channel canlead to large systematic errors in the inferred WIMP annihilation cross section. We demonstrate that ~ 66% of the CMSSM parameter space violates the equilibrium condition between capture and annihilation in the center of the Sun. By cross-correlating our predictions with direct detection methods, we conclude that if IceCube does detect a neutrino flux from the Sun at high significance while direct detection experiments do not find a signal above a spin-independent cross section sigma_SI^p larger than 5x10^{-9} pb, the CMSSM will be strongly disfavoured, given standard astrophysical assumptions for the WIMP distribution. This result is robust with respect to a change of priors. We argue that the proposed low-energy DeepCore extension of IceCube will be an ideal instrument to focus on relevant CMSSM areas of parameter space.
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