  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.
