Summary: This paper extends the iterative method of A. A. Abramov and A. N. Gaipova [Sov. Math., Dokl. 14, 1387–1390 (1973; Zbl 0293.47019); translation from Dokl. Akad. Nauk SSSR 212, 529–531 (1973)] which was designed for the solution of strongly monotone operator equations to the solution of set-valued strongly monotone variational inequalities in Hilbert space. The step-size rule in the presented projection method is based on a modified hypercircle estimate. Compared to other iteration processes this new iterative method does not require Lipschitz continuity of the monotone operator and nevertheless strong convergence with error bounds can be shown.