  Landauer's bound relates changes in the entropy of a system with the
inevitable dissipation of heat to the environment. The bound, however, becomes
trivial in the limit of zero temperature. Here we show that it is possible to
derive a tighter bound which remains non-trivial even as $T\to 0$. As in the
original case, the only assumption we make is that the environment is in a
thermal state. Nothing is said about the state of the system or the kind of
system-environment interaction. Our bound is valid for all temperatures and is
always tighter than the original one, tending to it in the limit of high
temperatures.
