A time-discrete formulation of the variational principle of mechanics is used to construct a novel first order; fixed time step integration method for multibody systems subject to mixed constraints. The new stepper; coined Spook; includes physics motivated constraint regularization and stabilization terms. The stepper is proved to be stable for the case of linear constraints; for non-zero regularization and stabilization parameters. For fixed stabilization value; the regularization can be made arbitrarily small; corresponding to arbitrarily stiff penalty forces. The â€śrelaxedâ€ť constraint formulation permits a separation of time scales so that stiff forces are treated as relaxed constraints. Constraint stabilization makes the stiff forces modeled this way strictly dissipative; and thus; the stepper essentially filters out the high oscillations; but is rigorously symplectic for the rest of the motion. Spook solves a single linear system per time step and is insensitive to constraint degeneracies for non-zero regularization. In addition; it keeps the constraint violations within bounds of O(h2); where h is the time step. Because it is derived from the discrete variational principle; the stepping scheme globally preserves the symmetries of the physical system. The combination of these features make Spook a very good choice for interactive simulations. Numerical experiments on simple multibody systems are presented to demonstrate the performance and stability properties.