The weak-localization contribution deltasigma(B) to the conductivity of a tunnel-coupled double-layer electron system is evaluated and its behavior in weak magnetic fields B perpendicular or parallel to the layers is examined. In a perpendicular field B, delta sigma(B) increases and remains dependent on tunneling as long as the magnetic field is smaller than hbar/e D tau_t, where D is the in-plane diffusion coefficient and tau_t the interlayer tunneling time. If tau_t is smaller than the inelastic scattering time, a parallel magnetic field also leads to a considerable increase of the concuctivity starting with a B**2 law and saturating at fields higher than hbar/e Z (D tau_t)**(1/2), where Z is the interlayer distance. In the limit of coherent tunneling, when tau_t is comparable to elastic scattering time, delta sigma(B) differs from that of a single-layer system due to ensuing modifications of the diffusion coefficient. A possibility to probe the weak-localization effect in double-layer systems by the dependence of the conductivity on the gate-controlled level splitting is discussed.
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