  We prove that a Hopf algebra of prime dimension $p$ over an algebraically
closed field, whose characteristic is equal to $p$, is either a group algebra
or a restricted universal enveloping algebra. Moreover, we show that any Hopf
algebra of prime dimension $p$ over a field of characteristic $q>0$ is
commutative and cocommutative when $q=2$ or $p<4q$. This problem remains open
in positive characteristic when $2<q<p/4$.
