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We develop a new method to constrain the star formation histories, dust attenuation and stellar masses of galaxies. It is based on two stellar absorption-line indices, the 4000-Å break strength and the Balmer absorption-line index Hδ_A. Together, these indices allow us to constrain the mean stellar ages of galaxies and the fractional stellar mass formed in bursts over the past few Gyr. A comparison with broad-band photometry then yields estimates of dust attenuation and of stellar mass. We generate a large library of Monte Carlo realizations of different star formation histories, including starbursts of varying strength and a range of metallicities. We use this library to generate median likelihood estimates of burst mass fractions, dust attenuation strengths, stellar masses and stellar mass-to-light ratios for a sample of 122 808 galaxies drawn from the Sloan Digital Sky Survey. The typical 95 per cent confidence range …