Pluto's Sputnik Planitia is a bright, roughly circular feature that resembles a polar ice cap. It is approximately 1,000 kilometres across and is centred on a latitude of 25 degrees north and a longitude of 175 degrees, almost directly opposite the side of Pluto that always faces Charon as a result of tidal locking. One explanation for its location includes the formation of a basin in a giant impact, with subsequent upwelling of a dense interior ocean. Once the basin was established, ice would naturally have accumulated there. Then, provided that the basin was a positive gravity anomaly (with or without the ocean), true polar wander could have moved the feature towards the Pluto-Charon tidal axis, on the far side of Pluto from Charon. Here we report modelling that shows that ice quickly accumulates on Pluto near latitudes of 30 degrees north and south, even in the absence of a basin, because, averaged over its orbital period, those are Pluto's coldest regions. Within a million years of Charon's formation, ice deposits on Pluto concentrate into a single cap centred near a latitude of 30 degrees, owing to the runaway albedo effect. This accumulation of ice causes a positive gravity signature that locks, as Pluto's rotation slows, to a longitude directly opposite Charon. Once locked, Charon raises a permanent tidal bulge on Pluto, which greatly enhances the gravity signature of the ice cap. Meanwhile, the weight of the ice in Sputnik Planitia causes the crust under it to slump, creating its own basin (as has happened on Earth in Greenland). Even if the feature is now a modest negative gravity anomaly, it remains locked in place because of the permanent tidal bulge raised by Charon. Any movement of the feature away from 30 degrees latitude is countered by the preferential recondensation of ices near the coldest extremities of the cap. Therefore, our modelling suggests that Sputnik Planitia formed shortly after Charon did and has been stable, albeit gradually losing volume, over the age of the Solar System.