We report the implementation and performance of a double servo-loop for intensity and phase-difference active stabilization of a dual-frequency vertical external--cavity surface-emitting laser (DF-VECSEL) for coherent population trapping (CPT) of cesium atoms in the framework of compact atomic clocks. In--phase fully correlated pumping of the two laser modes is identified as the best scheme for intensity noise reduction, and an analytical model allows the optimization of the active stabilization strategy. Optical phase-locking the beat-note to a local oscillator leads to a phase noise level below -103~dBc/Hz at 100~Hz from the carrier. The laser contribution to the short-term frequency stability of the clock is predicted to be compatible with a targeted Allan deviation below $sigma_y = 5,times 10^{-13}$ over one second.
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