  We present a Bayesian parameter-estimation pipeline to measure the properties
of inspiralling stellar-mass black hole binaries with LISA. Our strategy (i) is
based on the coherent analysis of the three noise-orthogonal LISA data streams,
(ii) employs accurate and computationally efficient post-Newtonian waveforms
accounting for both spin-precession and orbital eccentricity, and (iii) relies
on a nested sampling algorithm for the computation of model evidences and
posterior probability density functions of the full 17 parameters describing a
binary. We demonstrate the performance of this approach by analyzing the LISA
Data Challenge (LDC-1) dataset, consisting of 66 quasi-circular, spin-aligned
binaries with signal-to-noise ratios ranging from 3 to 14 and times to merger
ranging from 3000 to 2 years. We recover 22 binaries with signal-to-noise ratio
higher than 8. Their chirp masses are typically measured to better than $0.02
M_\odot$ at $90\%$ confidence, while the sky-location accuracy ranges from 1 to
100 square degrees. The mass ratio and the spin parameters can only be
constrained for sources that merge during the mission lifetime. In addition, we
report on the successful recovery of an eccentric, spin-precessing source at
signal-to-noise ratio 15 for which we can measure an eccentricity of $3\times
10^{-3}$.
