  The weak-localization contribution \delta\sigma(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.
