In this letter we investigate the one-layer coding/shaping system with single-level codes and sigmamapping for the bandwidth-limited regime. Specifically, we consider non-uniform memoryless sources sent over AWGN channels. At the transmitter, binary data are encoded by a Turbo code composed by two identical RSC (Recursive Systematic Convolutional) encoders. The encoded bits are randomly interleaved and modulated before entering the sigma-mapper. The modulation employed in this system follows the unequal energy allocation scheme first introduced in [?]. At each time instant, the sigma-mapper superimposes multiple modulated symbols so as to output an amplitude signal. The receiver consists of an iterative demapping/decoding algorithm, which incorporates the a priori probabilities of the source symbols. Simulation results show that a significant improvement is obtained when the unequal energy allocation scheme is used rather than a standard BPSK modulation. Furthermore, the proposed scheme is compared with that proposed in [?] for the power-limited regime and unequal energy allocation.