The ability to cool and manipulate atoms with light has yielded atom interferometry, precision spectroscopy, Bose-Einstein condensates and atom lasers. The extension of controlled manipulation to molecules is expected to be similarly rewarding, but molecules are not as amenable to manipulation by light owing to a far more complex energy-level spectrum. However, time-varying electric and magnetic fields have been successfully used to control the position and velocity of ions, suggesting that these schemes can also be used to manipulate neutral particles having an electric or magnetic dipole moment. Although the forces exerted on neutral species are many orders of magnitude smaller than those exerted on ions, beams of neutral dipolar molecules have been successfully slowed down in a series of pulsed electric fields and subsequently loaded into an electrostatic trap. Here we extend the scheme to include a prototype electrostatic storage ring made of a hexapole torus with a circumference of 80 cm. After injection, decelerated bunches of deuterated ammonia molecules, each containing about 106 molecules in a single quantum state and with a translational temperature of 10 mK, travel up to six times around the ring. Stochastic cooling might provide a means to increase the phase-space density of the stored molecules in the storage ring, and we expect this to open up new opportunities for molecular spectroscopy and studies of cold molecular collisions.