  The paper deals with an extremal problem concerning equitable colorings of
uniform hyper\-graph. Recall that a vertex coloring of a hypergraph $H$ is
called proper if there are no monochro-matic edges under this coloring. A
hypergraph is said to be equitably $r$-colorable if there is a proper coloring
with $r$ colors such that the sizes of any two color classes differ by at most
one. In the present paper we prove that if the number of edges $|E(H)|\leq
0.01\left(\frac{n}{\ln n}\right)^{\frac {r-1}{r}}r^{n-1}$ then the hypergraph
$H$ is equitably $r$-colorable provided $r<\sqrt[5]{\ln n}$.
