  Let $n\geq k\geq r+3$ and $\mathcal H$ be an $n$-vertex $r$-uniform
hypergraph. We show that if $|\mathcal H|> \frac{n-1}{k-2}\binom{k-1}{r}$ then
$\mathcal H$ contains a Berge cycle of length at least $k$. This bound is tight
when $k-2$ divides $n-1$. We also show that the bound is attained only for
connected $r$-uniform hypergraphs in which every block is the complete
hypergraph $K^{(r)}_{k-1}$. We conjecture that our bound also holds in the case
$k=r+2$, but the case of short cycles, $k\leq r+1$, is different.
