On varieties of associative algebras with weak growth

We prove that any variety of associative algebras with weak growth of the sequence $\{c_n(\mathbf{V})\}_{n\geq 1}$ satisfies the identity $[x_1,x_2][x_3,x_4]\ldots [x_{2s-1},x_{2s}]=0$ for some $s$. As a consequence, the exponent of an arbitrary associative variety with weak growth exists and is an integer and if the characteristic of the ground field is distinct from 2 then there exists no varieties of associative algebras whose growth is intermediate between polynomial and exponential.