We demonstrate a vapor-cell atomic-clock prototype based on a continuous-wave interrogation and double-modulation coherent population trapping (DM-CPT) technique. The DM-CPT technique uses a synchronous modulation of polarization and the relative phase of a bichromatic laser beam in order to increase the number of atoms trapped in a dark state, i.e., a nonabsorbing state. The narrow resonance, observed in the transmission of a Cs vapor cell, is used as a narrow frequency discriminator in an atomic clock. A detailed characterization of the CPT resonance versus numerous parameters is reported. A short-term fractional-frequency stability of 3.2×10−13τ−1/2 up to a 100-s averaging time is measured. These performances are more than one order of magnitude better than industrial Rb clocks and are comparable to those of the best laboratory-prototype vapor-cell clocks. The noise-budget analysis shows that the short- and midterm frequency stability is mainly limited by the power fluctuations of the microwave used to generate the bichromatic laser. These preliminary results demonstrate that the DM-CPT technique is well suited for the development of a high-performance atomic clock, with the potential compact and robust setup due to its linear architecture. This clock could find future applications in industry, telecommunications, instrumentation, or global navigation satellite systems.