In our recent papers [1, 2] we have shown how to find a regularization parameter such that the unique solution of a strongly monotone variational inequality can be approximated by the Banach contraction mapping principle. In this paper we combine this result with the proximal point algorithm to obtain a new projection-type algorithm for solving (not necessarily strongly) monotone variational inequalities. The proposed algorithm does not require knowing any Lipschitz constant of the cost operator. The main subproblem in the proposed algorithm is of computing the projection of a point onto a closed convex set. Application of the proposed algorithm to an equilibrium problem is discussed. Computational results are reported.