Radio networks for future mobile communications systems, for example, 3GPP Long-Term Evolution (LTE), are likely to use an orthogonal frequency division multiplexing- (OFDM-) based air interface in the downlink with a frequency reuse factor of one to avoid frequency planning. Therefore, system capacity is limited by interference, which is particularly crucial for mobile terminals with a single receive antenna. Nevertheless, next generation mobile communications systems aim at increasing downlink throughput. In this paper, a single antenna interference cancellation (SAIC) algorithm is introduced for amplitude-shift keying (ASK) modulation schemes in combination with bit-interleaved coded OFDM. By using such a transmission strategy, high gains in comparison to a conventional OFDM transmission with quadrature amplitude modulation (QAM) can be achieved. The superior performance of the novel scheme is confirmed by an analytical bit-error probability (BEP) analysis of the SAIC receiver for a single interferer, Rayleigh fading, and uncoded transmission. For the practically more relevant multiple interferer case we present an adaptive least-mean-square (LMS) and an adaptive recursive least-squares (RLS) SAIC algorithm. We show that in particular the RLS approach enables a good tradeoff between performance and complexity and is robust even to multiple interferers.