  Gravitational memory, a residual change, arises after a finite gravitational
wave pulse interacts with free masses. We calculate the memory effect in
massive gravity as a function of the graviton mass $(m_g)$ and show that it is
discretely different from the result of general relativity: the memory is
reduced not just via the usual expected Yukawa decay but by a numerical factor
which survives even in the massless limit. For the strongest existing bounds on
the graviton mass, the memory is essentially wiped out for the sources located
at distances above 10 Mpc. On the other hand, for the weaker bounds found in
the LIGO observations, the memory is reduced to zero for distances above 0.1
Pc. Hence, we suggest that careful observations of the gravitational wave
memory effect can rule out the graviton mass or significantly bound it. We also
show that adding higher curvature terms reduces the memory effect.
