Long- and short-lived radioactive isotopes and their daughter products in meteorites are chronometers that can test models for Solar System formation. Differentiated meteorites come from parent bodies that were once molten and separated into metal cores and silicate mantles. Mineral ages for these meteorites, however, are typically younger than age constraints for planetesimal differentiation. Such young ages indicate that the energy required to melt their parent bodies could not have come from the most likely heat source-radioactive decay of short-lived nuclides ((26)Al and (60)Fe) injected from a nearby supernova-because these would have largely decayed by the time of melting. Here we report an age of 4.5662 +/- 0.0001 billion years (based on Pb-Pb dating) for basaltic angrites, which is only 1 Myr younger than the currently accepted minimum age of the Solar System and corresponds to a time when (26)Al and (60)Fe decay could have triggered planetesimal melting. Small (26)Mg excesses in bulk angrite samples confirm that (26)Al decay contributed to the melting of their parent body. These results indicate that the accretion of differentiated planetesimals pre-dated that of undifferentiated planetesimals, and reveals the minimum Solar System age to be 4.5695 +/- 0.0002 billion years.