  An exact algorithm is used to compute the degeneracies of the excited states
of the bimodal Ising spin glass in two dimensions. It is found that the
specific heat at arbitrary low temperature is not a self-averaging quantity and
has a distribution that is neither normal or lognormal. Nevertheless, it is
possible to estimate the most likely value and this is found to scale as L^3
T^(-2) exp(-4J/kT), for a L*L lattice. Our analysis also explains, for the
first time, why a correlation length \xi ~ exp(2J/kT) is consistent with an
energy gap of 2J. Our method allows us to obtain results for up to 10^5
disorder realizations with L <= 64. Distributions of second and third
excitations are also shown.
