Low-mass white-dwarf stars are the remnants of disrupted red-giant stars in binary millisecond pulsars(1) and other exotic binary star systems(2-4). Some low-mass white dwarfs cool rapidly, whereas others stay bright for millions of years because of stable fusion in thick surface hydrogen layers(5). This dichotomy is not well understood, so the potential use of low-mass white dwarfs as independent clocks with which to test the spin-down ages of pulsars(6,7) or as probes of the extreme environments in which low-mass white dwarfs form(8-10) cannot fully be exploited. Here we report precise mass and radius measurements for the precursor to a low-mass white dwarf. We find that only models in which this disrupted red-giant star has a thick hydrogen envelope can match the strong constraints provided by our data. Very cool low-mass white dwarfs must therefore have lost their thick hydrogen envelopes by irradiation from pulsar companions(11,12) or by episodes of unstable hydrogen fusion (shell flashes). We also find that this low-mass white-dwarf precursor is a type of pulsating star not hitherto seen. The observed pulsation frequencies are sensitive to internal processes that determine whether this star will undergo shell flashes.