Let $(X,\mathcal A,\mu,T)$ be an ergodic invertible probability-preserving system. Let $p=(p_i)_{i\in I}$ be a finite or countable probability vector with Shannon entropy \begin{align*} H(p):=-\sum_{i\in I}p_i\log p_i=h, \end{align*} where $0<h\le h_\mu(T)$. Then $T$ has a Bernoulli factor isomorphic to the Bernoulli shift with base distribution $p$, and this factor has Kolmogorov-Sinai entropy exactly $h$.