The two-dimensional random bond Ising model with two distinct ferromagnetic nearest-neighbour couplings, J and J', distributed randomly over the lattice with concentration 1/2 is studied for various ratios J'/J at criticality. The critical temperature is exactly known from self-duality. The size-dependent specific heat shows a crossover from the perfect Ising to the randomness-dominated critical behaviour of doubly logarithmic form. In the latter region, our estimates for the critical exponents of the size-dependent magnetization and susceptibility are in very good agreement with analytic results suggesting β/ν = 1/8 and γ/ν = 7/4.