  The energy-energy correlation function of the two-dimensional Ising model
with weakly fluctuating random bonds is evaluated in the large scale limit. Two
correlation lengths exist in contrast to one correlation length in the pure 2D
Ising model: one is finite whereas the other is divergent at the critical
points. The corresponding exponent of the divergent correlation length is
$\nu_e=1/2$ in contrast to the pure system where $\nu_e=1$. The calculation is
based on a previously developed effective field theory for the energy density
fluctuations.
