Dust emission around the nearby star epsilon Eridani has been imaged using a new submillimetre camera (SCUBA at the JCMT). At 850 microns wavelength a ring of dust is seen, peaking at 60 AU from the star and with much lower emission inside 30 AU. The mass of the ring is at least 0.01 Earth masses in dust, while an upper limit of 0.4 Earth masses in molecular gas is imposed by CO observations. The total mass is comparable to the estimated amount of material, 0.04-0.3 Earth masses, in comets orbiting the Solar System. The most probable origin of the the ring structure is that it is a young analogue to the Kuiper Belt in our Solar System, and that the central region has been partially cleared by the formation of grains into planetesimals. Dust clearing around epsilon Eri is seen within the radius of Neptunes orbit, and the peak emission at 35-75 AU lies within the estimated Kuiper Belt zone of 30-100 AU radius. epsilon Eri is a main-sequence star of type K2V (0.8 Solar masses) with an estimated age of 0.5-1.0 Gyr, so this interpretation is consistent with the early history of the Solar System where heavy bombardment occurred up to approximately 0.6 Gyr. An unexpected discovery is substructure within the ring, and these asymmetries could be due to perturbations by planets.