The main result of the paper is a characterization of weak congruence modular varieties (every algebra of which has a modular lattice of weak congruences). Varieties are supposed to have a nullary operation, and every algebra a one element subalgebra. It. is proved that such a variety is weak congruence modular if and only if it is polynomially equivalent to the variety of modules over a ring with unit. Some other characterizations of such varieties and of algebras in these varieties having distributive weak congruence lattices, are also given.