This paper introduces an algorithm for the registration of rotated and translated volumes using the three-dimensional (3-D) pseudopolar Fourier transform, which accurately computes the Fourier transform of the registered volumes on a near-spherical 3-D domain without using interpolation. We propose a three-step procedure. The first step estimates the rotation axis. The second step computes the planar rotation relative to the rotation axis. The third step recovers the translational displacement. The rotation estimation is based on Euler's theorem, which allows one to represent a 3-D rotation as a planar rotation around a 3-D rotation axis. This axis is accurately recovered by the 3-D pseudopolar Fourier transform using radial integrations. The residual planar rotation is computed by an extension of the angular difference function [1] to cylindrical motion. Experimental results show that the algorithm is accurate and robust to noise.