  We compute the statistical distribution of the transmittance of a random
waveguide with absorption in the limit of many propagating channels. We
consider the average and fluctuations of the conductance T = tr t^{\dagger} t,
where t is the transmission matrix, the density of transmission eigenvalues
\tau (the eigenvalues of t^{\dagger} t), and the distribution of the plane-wave
transmittances T_a and T_{ab}. For weak absorption (length L smaller than the
exponential absorption length \xi_a), we compute moments of the distributions,
while for strong absorption (L >> \xi_a), we can find the complete
distributions. Our findings explain recent experiments on the transmittance of
random waveguides by Stoytchev and Genack [Phys. Rev. Lett. 79, 309 (1997)].
