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The smallest network model that captures relaying in the presence of multiple communicating pairs causing interference to each other is the interference channel with a relay. In this paper, an achievable rate region for the interference channel with a relay is derived. Special cases of strong interference under which this region is the capacity region are presented. The results obtained demonstrate the benefits of interference forwarding at a relay. By forwarding interfering messages, the relay can improve their reception at unintended receivers and, thus, facilitate interference cancellation. We show that intentionally forwarding interfering messages can improve the achievable rates. The achievable rates and interference forwarding gains are also illustrated by numerical results in Gaussian channels. Finally, a sum-rate outer bound to the capacity region of the Gaussian interference channel with a relay is derived and compared with the achievable rate region. The cut-set bound for this channel is also derived and shown to be much looser than the new sum-rate outer bound.