  We derive high-temperature series expansions for the free energy and
susceptibility of the two-dimensional random-bond Ising model with a symmetric
bimodal distribution of two positive coupling strengths J_1 and J_2 and study
the influence of the quenched, random bond-disorder on the critical behavior of
the model. By analysing the series expansions over a wide range of coupling
ratios J_2/J_1, covering the crossover from weak to strong disorder, we obtain
for the susceptibility with two different methods compelling evidence for a
singularity of the form $\chi \sim t^{-7/4} |\ln t|^{7/8}$, as predicted
theoretically by Shalaev, Shankar, and Ludwig. For the specific heat our
results are less convincing, but still compatible with the theoretically
predicted log-log singularity.
