  Gregory and Laflamme showed that certain nonextremal black strings (and
p-branes) are unstable to linearized perturbations. It is widely believed that
this instability will cause the black string horizon to classically pinch off
and then quantum mechanically separate, resulting in higher dimensional black
holes. We argue that this cannot happen. Under very mild assumptions, classical
event horizons cannot pinch off. Instead, they settle down to new static black
string solutions which are not translationally invariant along the string.
