This paper addresses the design of constant-modulus waveforms for a dual-function multiple-input-multiple-output (MIMO) radar-communication system. The purpose of the design is to match a desired beampattern for radar sensing, and minimize the transmission distortion of communication signals. To tackle the non-convex waveform design problem, we develop an iterative algorithm based on cyclic optimization and majorization-minimization (MM). We show that the proposed algorithm has guaranteed convergence of the objective values. Numerical results demonstrate that the transmit beampattern of the synthesized waveforms can well approximate the desired one, and the emitted communication signals from the dual-function system has little distortions.