Observations over the last decade have revealed the existence of a large number of bodies orbiting the Sun beyond Neptune. Known as the Kuiper-belt objects (KBOs), they are believed to be formed in the outer reaches of the protoplanetary disk around the young Sun, and have been little altered since then. They are probably the source of short-period comets. The KBOs are, however, difficult objects to study because of their distance from earth, so even basic physical properties such as their sizes and albedos remain unknown. Previous size estimates came from assuming an albedo with the canonical value being 0.04. Here we report simultaneous measurements of the thermal emission and reflected optical light of the bright KBO (20000) Varuna, which allow us to determine independently both the size and the albedo. Varuna has an equivalent circular diameter of D = 900+129-145 km and a red geometric albedo of pR = 0.070+0.030-0.017. Its surface is darker than Pluto's, suggesting that it is largely devoid of fresh ice, but brighter than previously assumed for KBOs.