  We study the dynamics of supermassive black hole binaries embedded in
circumbinary gaseous discs, with the SPH code Gadget-2. The sub-parsec binary
(of total mass M and mass ratio q=1/3) has excavated a gap and transfers its
angular momentum to the self--gravitating disc (M_disc=0.2 M). We explore the
changes of the binary eccentricity e, by simulating a sequence of binary models
that differ in the initial eccentricity e_0, only. In initially low-eccentric
binaries, the eccentricity increases with time, while in high-eccentric
binaries e declines, indicating the existence of a limiting eccentricity e_crit
that is found to fall in the interval [0.6,0.8]. We also present an analytical
interpretation for this saturation limit. An important consequence of the
existence of e_crit is the detectability of a significant residual eccentricity
e_LISA} by the proposed gravitational wave detector LISA. It is found that at
the moment of entering the LISA frequency domain e_LISA ~ 10^{-3}-10^{-2}; a
signature of its earlier coupling with the massive circumbinary disc. We also
observe large periodic inflows across the gap, occurring on the binary and disc
dynamical time scales rather than on the viscous time. These periodic changes
in the accretion rate (with amplitudes up to ~100%, depending on the binary
eccentricity) can be considered a fingerprint of eccentric sub-parsec binaries
migrating inside a circumbinary disc.
