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We apply low-density parity-check (LDPC) codes to a bandwidth-efficient modulation scheme using multilevel coding, multistage decoding, and trellis-based signal shaping. Performance results based on density evolution and simulations are presented. Using irregular LDPC component codes of block length 105 and a 64-quadrature amplitude modulation signal constellation operating at 2 bits/dimension, a bit-error rate of 10-5 is achieved at an Eb/N0 of 6.55 dB. At this value of Eb/N0, the Shannon channel capacity, computed assuming equally likely signaling, is below 2 bits/dimension.