'Cataclysmic variables' are binary star systems in which one star of the pair is a white dwarf, and which often generate bright and energetic stellar outbursts. Classical novae are one type of outburst: when the white dwarf accretes enough matter from its companion, the resulting hydrogen-rich atmospheric envelope can host a runaway thermonuclear reaction that generates a rapid brightening(1-4). Achieving peak luminosities of up to one million times that of the Sun(5), all classical novae are recurrent, on timescales of months(6) to millennia(7). During the century before and after an eruption, the 'novalike' binary systems that give rise to classical novae exhibit high rates of mass transfer to their white dwarfs(8). Another type of outburst is the dwarf nova: these occur in binaries that have stellar masses and periods indistinguishable from those of novalikes(9) but much lower mass-transfer rates(10), when accretion-disk instabilities(11) drop matter onto the white dwarfs. The coexistence at the same orbital period of novalike binaries and dwarf novae-which are identical but for their widely varying accretion rates-has been a longstanding puzzle(9). Here we report the recovery of the binary star underlying the classical nova eruption of 11 March ad 1437 (refs 12, 13), and independently confirm its age by proper-motion dating. We show that, almost 500 years after a classical-nova event, the system exhibited dwarf-nova eruptions. The three other oldest recovered classical novae(14-16) display nova shells, but lack firm post-eruption ages(17,18), and are also dwarf novae at present. We conclude that many old novae become dwarf novae for part of the millennia between successive nova eruptions(19,20).