We present a variant of Korpelevich's method for variational inequality problems with monotone operators. Instead of a fixed and exogenously given stepsize, possible only when a Lipschitz constant for the operator exists and is known beforehand, we find an appropriate stepsize in each iteration through an Armijo-type search. Differently from other similar schemes, we perform only two projections onto the feasible set in each iteration, rather than one projection for each tentative step during the search, which represents a considerable saving when the projection is computationally expensive. A full convergence analysis is given, without any Lipschitz continuity assumption