Standard time integration schemes such as Newmark’s method are well known to have stability issues and chatter in the event of unilateral contact. This is due to inherent smoothness built into these time integration schemes and the lack of control of the energy due to the nonlinear nature of the impact. A large body of literature has been developed to treat these issues and a small survey of the different approaches will be presented. A number of approaches apply Lagrange multipliers and employ a velocity jump (e.g. [1, 2]) in the time integration scheme to handle impulse occuring at impact. Some employ incremental gap forms [3, 2] whilst others apply special techniques to enforce the explicit gap constraint at each time step [4, 5]. Some employ penalty constraints and balance energy at contact release [3]. Several incorporate special schemes to add algorithmic damping to mitigate high frequency noise [4]. This work combines different aspects from previous approaches, includes Lagrange multipliers and yields an algorithm that is provably B-stable in a discrete energy norm. Here, energy is dissipated at impact and the velocity is updated by applying a bilateral constraint to enforce persistency. This process is inherently dissipative but an additional force can be applied to recover the lost energy upon contact release if desired. In fact, the proposed conserving approach only requires two subsequent linear solves at the end of time step to recover lost energy as opposed to the nonlinear solve necessary in [2]. The time integration scheme is applied to a modified mortar contact formulation and is not only provably stable in time but also exactly conserves discrete forms of linear and angular momentum.