  Binary black hole (BBH) mergers are the primary sources of gravitational wave
(GW) events detected by LIGO/Virgo. Binary black holes embedded in the
accretion discs of active galactic nuclei (AGN) are possible candidates for
such GW events. We develop an idealised analytic model for the orbital
evolution of BBHs in AGN accretion discs by combining the evolution equations
of disc-binary interaction and GW inspiral. We investigate the coupled
`disc+GW'-driven evolution of BBHs transitioning from the disc-driven regime at
large orbital separations into the GW-driven regime at small separations. In
this evolution channel, BBH mergers are accelerated by a combination of orbital
decay and orbital eccentricity growth in the disc-dominated regime. We provide
a quantification of the resulting merger timescale $\tau_\text{merger}$, and
analyse its dependence on both the accretion disc and binary orbital
parameters. By computing the evolution of the orbital eccentricity as a
function of the GW frequency, we predict that most binaries in AGN discs should
have significant residual eccentricities ($e \sim 0.01-0.1$), potentially
detectable by LISA. We further discuss the potentials and caveats of this
particular BBH-in-AGN channel in the framework of binary evolutionary paths.
