We find the complete and explicit general solution of the conformal Killing equation in static spherically symmetric spacetimes, thus unifying and generalizing previous special cases. For non-conformally-flat spacetimes, there are at most two proper conformal motions. There are three classes of such spacetimes, and one or both of the conformal Killing vectors is non-inheriting. One of the classes includes self-similar spacetimes (i.e. with a homothetic motion). The conformally flat spacetimes (including the Schwarzschild interior metric) fall into three classes, and their eleven proper conformal Killing vectors are given in full. The only spacetimes with conformal motion that are regular at the centre are conformally flat. An addendum for this article has been published in 1996 Class. Quantum Grav. 13 317.