Summary: An algorithm for computing a stationary point of a quadratic program with box constraints (BQP) is proposed. Each iteration of this procedure comprises a guessing strategy which forecasts the active bounds at a stationary point, the determination of a descent direction by means of solving a reduced strictly convex quadratic program with box constraints and an exact line search. Global convergence is established in the sense that every accumulation point is stationary. Moreover, it is shown that the algorithm terminates after a finite number of iterations, if at least one iterate is sufficiently close to a stationary point which satisfies a certain sufficient optimality condition. The algorithm can be easily implemented for sparse large-scale BQPs. Furthermore, it simplifies for concave BQPs, as it is not required to solve strictly convex quadratic programs in this case. Computational experience with large-scale BQPs is included and shows the appropriateness of this type of methodology.