Increased demand for wireless services has spurred research into spectrum co-existence between radar and communication systems. The effect of an unaltered radar interference on an uncoded communication receiver has been investigated for constellations used in commercial systems. However, the constellation design — possible for a cognitive transmitter-receiver pair — that maximizes the transmission rate for communication systems affected by Gaussian noise and radar interference has not yet been addressed and is tackled here. Two regimes are investigated: when the Interference-to-Noise Ratio (INR) of the radar signal is very low and very high compared to the Signal-to-Noise Ratio (SNR) of the communication signal. It is found that: (a) When INR≪SNR, the channel behaves as a complex-valued AWGN channel; the designed constellations exhibit a concentric (almost equilateral) hexagonal structure as the number of signal points increases, (b) When INR≫SNR≫1, the channel behaves as a real-valued phase-fading AWGN channel; the designed constellations are shaped as an (almost equally-spaced) pulse-amplitude modulation as the number of signal points increases, and (c) The designed constellations outperform practically-used ones in terms of the transmission rate.