  We construct dark solitons in the recently introduced model of the nonlinear
dual-core coupler with the mutually balanced gain and loss applied to the two
cores, which is a realization of parity-time symmetry in nonlinear optics. The
main issue is stability of the dark solitons. The modulational stability of the
CW (continuous-wave) background, which supports the dark solitons, is studied
analytically, and the full stability is investigated in a numerical form, via
computation of eigenvalues for modes of small perturbations. Stability regions
are thus identified in the parameter space of the system, and verified in
direct simulations. Collisions between stable dark solitons are briefly
considered too.
