  We investigate the one-dimensional quantum $XXZ$ model in the presence of
diagonal disorder. Recently the model has been analyzed with the help of
field-theoretical renormalization group methods, and a phase diagram has been
predicted. We study the model with exact diagonalization techniques up to chain
lengths of 16 sites. Using finite-size scaling methods we estimate critical
exponents and the phase diagram and find reasonably good agreement with the
field-theoretical results, namely, that any amount of disorder destroys the
superfluidity for $XXZ$ anisotropy $\Delta$ between $ -1/2 $ and $1$, while the
superfluidity persists to finite disorder strength for $ -1 < \Delta < -1/2$
and then undergoes a Kosterlitz-Thouless type transition.
