The soft-tissue imaging capabilities of magnetic resonance imaging (MRI) combined with high precision robotics has the potential to improve the precision and safety of a wide range of image-guided medical procedures. However, functional MRI-compatible robotics have not yet been realized in part because conventional electromagnetic servomotors can become dangerous projectiles near the strong magnetic field of an MRI scanner. Here we report an electromagnetic servomotor constructed from non-magnetic components, where high-torque and controlled rotary actuation is produced via interaction between electrical current in the servomotor armature and the magnetic field generated by the superconducting magnet of the MRI scanner itself. Using this servomotor design, we then build and test an MRI-compatible robot which can achieve the linear forces required to insert a large-diameter biopsy instrument in tissue during simultaneous MRI. Our electromagnetic servomotor can be safely operated (while imaging) in the patient area of a 3 Tesla clinical MRI scanner.