  Binary systems of massive black holes will be detectable by the Laser
Interferometer Space Antenna (LISA) throughout the entire Universe.
Observations of gravitational waves from this class of sources will have
important repercussions on our understanding of the behaviour of gravity in the
highly non-linear relativistic regime, the distribution and interaction of
massive black holes at high redshift and the formation and evolution of cosmic
structures. It is therefore important to address how accurately LISA can
measure the source parameters and explore the implications for astronomy and
cosmology. Present observations and theoretical models suggest that massive
black holes could be spinning, possibly rapidly in some cases. In binary
systems, the relativistic spin-orbit interaction causes the orbital plane to
precess in space producing a characteristic signature on the emitted
gravitational waves. In this paper we investigate the effect of spins on the
gravitational wave signal registered at the LISA output and the implications
for parameter estimation. We consider the in-spiral phase of binary systems in
circular orbit undergoing the so-called "simple precession" and we approximate
the gravitational radiation at the restricted 1.5PN order. We show that the
presence of spins changes dramatically the signature of the signal recorded by
LISA. As a consequence, the mean square errors associated to the parameter
measurements are significantly smaller than the ones obtained when the effect
of spins is neglected. For a binary system of two $10^6 \Ms$ black holes, the
angular resolution and the relative error on the luminosity distance improve by
a factor $\approx$ 3-to-10; the fractional errors on the chirp mass and the
reduced mass decrease by a factor $\sim 10$ and $\sim 10^3$, respectively.
