Gravitational waves from the inspiral and coalescence of supermassive black-hole (SMBH) binaries with masses m₁ ~ m₂ ~ 10⁶M are likely to be among the strongest sources for the Laser Interferometer Space Antenna (LISA). We describe a three-stage data-analysis pipeline designed to search for and measure the parameters of SMBH binaries in LISA data. The first stage uses a time–frequency track-search method to search for inspiral signals and provide a coarse estimate of the black-hole masses m₁, m₂ and the coalescence time of the binary t_c. The second stage uses a sequence of matched-filter template banks, seeded by the first stage, to improve the measurement accuracy of the masses and coalescence time. Finally, a Markov chain Monte Carlo search is used to estimate all nine physical parameters of the binary (masses, coalescence time, distance, initial phase, sky position and orientation). Using results from the second stage substantially shortens the Markov chain burn-in time and allows us to determine the number of SMBH-binary signals in the data before starting parameter estimation. We demonstrate our analysis pipeline using simulated data from the first Mock LISA Data Challenge. We discuss our plan for improving this pipeline and the challenges that will be faced in real LISA data analysis.