  We investigate the evolution of a massive black hole pair under the action of
dynamical friction (DF) by a uniform background of light stars with isotropic
velocity distribution. In our scenario, the primary black hole $M_1$ sits, at
rest, in the center of the spherical star distribution (with mass $M_c$) and
the secondary less massive companion $M_2$ moves along bound orbits determined
by the background gravitational field; it loses energy $E$ and angular momentum
$J$ by DF. We investigate mostly analytically the secular evolution of the
orbital parameters when the motion of $M_2$ is determined either by the mean
field generated by the uniform stellar distribution or by the gravitational
field of $M_1$. We find that $J$ and $E$ are lost so as to cause the increase
of the eccentricity $e$ during the orbital decay of $M_2$. When $M_2$ enters
the region where the gravitational field of $M_1$ dominates, the evolution
depends on the ratio between the black hole velocity $v$ and the stellar
dispersion velocity $\sigma.$ We explore both cases $v<\sigma$ and $v>\sigma$.
Only for $v>\sigma$ the rise of $e$ would be severe but this transition occurs
close to the cusp radius below which our description becomes invalid. Energy
losses by gravitational wave emission become comparable to those by dynamical
friction at a critical distance that depends sensitively on the ratio
$M_1/M_c$. The braking index $n$ is calculated in this transition region.
