Let  ${T_t}_{t\ge 0}$ be a hypercyclic strongly continuous semigroup of operators. Then each  operator $T_t$ is hypercyclic as a single operator, and it shares the set of hypercyclic vectors with the semigroup. This answers in the affirmative a natural question concerning hypercyclic C0-semigroups. The analogous result for frequent hypercyclicity is also obtained. (This result is also known as the Conejero-Müller-Peris theorem in linear dynamics and generalizes and complete a previous result of Oxtoby and Ulam in Annals of Mathematics from 1941.