The power and bandwidth efficient constant-envelope BPSK (CE-BPSK) modulation is proposed. The CE-BPSK signal is realized specifying the phase transition characteristics for the conventional low pass filtered BPSK signal. Since the CE-BPSK signal has constant envelope and modified waveform, the CE-BPSK signal has better power and bandwidth efficiency compared to the conventional BPSK signal while the CE-BPSK signal is backward compatible to the conventional BPSK signal. It is also shown that the bit error rate performance of the CE-BPSK signal is the same as that of the conventional BPSK signal.