The formation of planets around binary stars may be more difficult than around single stars(1-3). In a close binary star (with a separation of less than a hundred astronomical units), theory predicts the presence of circumstellar disks around each star, and an outer circum-binary disk surrounding a gravitationally cleared inner cavity around the stars(4,5). Given that the inner disks are depleted by accretion onto the stars on timescales of a few thousand years, any replenishing material must be transferred from the outer reservoir to fuel planet formation (which occurs on timescales of about one million years). Gas flowing through disk cavities has been detected in single star systems(6). A circumbinary disk was discovered around the young low-mass binary system GG Tau A(ref. 7), which has recently been shown to be a hierarchical triple system(8). It has one large inner disk(9) around the single star, GG Tau Aa, and shows small amounts of shocked hydrogen gas residing within the central cavity(10), but other than a single weak detection(11), the distribution of cold gas in this cavity or in any other binary or multiple star system has not hitherto been determined. Here we report imaging of gas fragments emitting radiation characteristic of carbon monoxide within the GG Tau A cavity. From the kinematics we conclude that the flow appears capable of sustaining the inner disk (around GG Tau Aa) beyond the accretion lifetime, leaving time for planet formation to occur there. These results show the complexity of planet formation around multiple stars and confirm the general picture predicted by numerical simulations.