In many gravitational interactions between galaxies, gas and stars that have been torn from either or both of the precursor galaxies can collect in \'tidal tails\'. Star formation begins anew in these regions to produce \'tidal dwarf galaxies\', giving insight into the process of galaxy formation through the well-defined timescale of the interaction. But tracking the star formation process has proved to be difficult: the tidal dwarf galaxies with young stars showed no evidence of the molecular gas out of which new stars form. Here we report the discovery of molecular gas (carbon monoxide emission) in two tidal dwarf galaxies. In both cases, the molecular gas peaks at the same location as the maximum in atomic-hydrogen density, unlike most gas-rich galaxies. We infer from this that the molecular gas formed from the HI, rather than being torn in molecular form from the interacting galaxies. Star formation in the tidal dwarfs appears to mimic that process in normal spiral galaxies like our own.