The size and limited dexterity of current surgical robotic systems are factors which limit their usefulness. To improve the level of assimilation of surgical robots in minimally invasive surgery (MIS), a compact, lightweight surgical robotic positioning mechanism with four degrees of freedom (DOF) (three rotational DOF and one translation DOF) is proposed in this paper. This spatial mechanism based on a bevel-gear wrist is remotely driven with three rotation axes intersecting at a remote rotation center (the MIS entry port). Forward and inverse kinematics are derived, and these are used for optimizing the mechanism structure given workspace requirements. By evaluating different spherical geared configurations with various link angles and pitch angles, an optimal design is achieved which performs surgical tool positioning throughout the desired kinematic workspace while occupying a small space bounded by a hemisphere of radius 13.7 cm. This optimized workspace conservatively accounts for collision avoidance between patient and robot or internally between the robot links. This resultant mechanism is highly compact and yet has the dexterity to cover the extended workspace typically required in telesurgery. It can also be used for tool tracking and skills assessment. Due to the linear nature of the gearing relationships, it may also be well suited for implementing force feedback for telesurgery.