Miniature atomic clocks based on coherent population trapping (CPT) states in thermal atoms are an important component in many field applications, particularly where satellite frequency standards are not accessible. Cold-atom CPT clocks promise improved accuracy and stability over existing commercial technologies. Here we demonstrate a cold-atom CPT clock based on ${}^{87}$Rb using a high-contrast double-$\ensuremath{\Lambda}$ configuration. Doppler frequency shifts are explained using a simple model and canceled by interrogating the atoms with counterpropagating light beams. We realize a compact cold-atom CPT clock with a fractional frequency stability of $4\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}11}{\ensuremath{\tau}}^{\ensuremath{-}1/2}$, thus demonstrating the potential of these devices. We also show that the long-term stability is currently limited by the second-order Zeeman shift to $2\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}12}$ at 1000 s.