  We study the inspiral of double black holes, with masses in the LISA window
of detectability, orbiting inside a massive circum-nuclear disc. Using
high-resolution SPH simulations, we follow the black hole dynamics in the early
phase when gas-dynamical friction acts on the black holes individually, and
continue our simulation until they form a close binary. We find that in the
early sinking the black holes lose memory of their initial orbital eccentricity
if they co-rotate with the gaseous disc, forming a binary with a low
eccentricity, consistent with zero within our numerical resolution limit. The
cause of circularization resides in the rotation present in the gaseous
background where dynamical friction operates. Circularization may hinder
gravitational waves from taking over and leading the binary to coalescence. In
the case of counter-rotating orbits the initial eccentricity does not decrease,
and the black holes may bind forming an eccentric binary. When dynamical
friction has subsided, for equal mass black holes and regardless their initial
eccentricity, angular momentum loss, driven by the gravitational torque exerted
on the binary by surrounding gas, is nevertheless observable down to the
smallest scale probed. In the case of unequal masses, dynamical friction
remains efficient down to our resolution limit, and there is no sign of
formation of any ellipsoidal gas distribution that may further harden the
binary. During inspiral, gravitational capture of gas by the black holes occurs
mainly along circular orbits: eccentric orbits imply high relative velocities
and weak gravitational focusing. Thus, AGN activity may be excited during the
black hole pairing process and double active nuclei may form when
circularization is completed, on distance-scales of tens of pcs.
