  We show how to extend the usual black string instability of vacuum or charged
black p-branes to the anti-de Sitter background. The string fragments in an
analogous fashion to the $\Lambda=0$ case, the main difference being that
instead of a periodic array of black holes forming, an accumulation of ``mini''
black holes occurs towards the AdS horizon. In the case where the AdS space is
of finite extent, such as an orbifold compactification, we show how the
instability switches off below a certain compactification scale.
