We report an exact transparent boundary condition (TBC) on the surface of a rectangular cuboid for the three-dimensional (3D) time-dependent Schrödinger equation. It is obtained as a generalization of the well-known TBC for the 1D Schrödinger equation and of the exact TBC in the rectangular domain for the 3D parabolic wave equation, which we reported earlier. Like all other TBCs, it is nonlocal in time domain and relates the boundary transverse derivative of the wave function at any given time to the boundary values of the same wave function at all preceding times. We develop a discretization of this boundary condition for the implicit Crank-Nicolson finite difference scheme. Several numerical experiments demonstrate evolution of the wave function in free space as well as propagation through a number of 3D spherically symmetrical and asymmetrical barriers, and, finally, scattering off an asymmetrical 3D potential. The proposed boundary condition is simple and robust, and can be useful in computational quantum mechanics when an accurate numerical solution of the 3D Schrödinger equation is required.