The one-dimensional quantum Heisenberg model with random +/-J bonds is
studied for S=1/2 and S=1. The specific heat and the zero-held suscep
tibility are calculated by using high-temperature series expansions an
d the quantum transfer matrix method. The susceptibility shows a Curie
-like temperature dependence at low temperatures as well as at high te
mperatures. The numerical results for the specific heat suggest that t
here are anomalously many low-lying excitations. The qualitative natur
e of these excitations is discussed based on the exact diagonalization
of finite-size systems.