In multiple array radar systems, waveform diversity can be exploited to enhance the beampattern design. However, in a spectrally crowded environment, the spectral shape of radar waveforms must be compatible with civilian and other wireless communication services. We address the problem of designing a beampattern for MIMO radar under a spectral interference constraint. The problem essentially reduces to a constrained design of the radar waveform, where a key open challenge is jointly enforcing the spectral interference constraint in addition to the constant modulus constraint (CMC). A new approach is proposed in our work, which involves solving a sequence of convex Constrained Quadratic Programs, each of which has a closed form solution and such that constant modulus is achieved at convergence. New analytical results are provided in support of the proposed Beampattern design under constant modulus and Interference Constraints (BIC). We evaluate the proposed BIC algorithm against state of the art MIMO beampattern design methods that enforce constant modulus and show that BIC achieves a high fidelity radar beampattern while maintaining low spectral interference in the desired bands.