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In this paper a coded modulation method for the additive white Gaussian noise channel is presented which incorporates signal shaping. The scheme uses a modulation map to map bits to channel inputs. A modulation map is considered that has the advantage it is can easily be combined with binary error-correcting codes. The decoding strategy is a combination of bit-interleaved coded modulation and multistage decoding. The number of required decoding stages is only small compared to other multistage decoding schemes for coded modulation. It is shown that modulation maps can be designed with a constrained capacity limit very close to the capacity of the AWGN channel. When combined with low-density parity-check codes, simulation results show that a near-capacity performance is achieved for high spectral efficiencies.