Let {T_t ; t>=0} be a hypercyclic strongly continuous semigroup of operators. Then each T_t (t > 0) 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 C_0-semigroups. The analogous result for frequent hypercyclicity is also obtained.