  For the Bergman projection operator $P$ we prove that $
\|P\|_{{L^1(B,d\lambda)\rightarrow B_1}}= \frac {(2n+1)!}{n!}.$ Here $\lambda$
stands for the invariant metric in the unit ball $B$ of $\mathbf{C}^n$, and
$B_1$ denotes the Besov space with an adequate semi--norm. We also consider a
generalization of this result. This generalizes some recent results due to
Per\"{a}l\"{a}.
