A single indium ion stored in a radio frequency trap and laser-cooled to a temperature below 1 mK can serve as an optical frequency standard of exceptionally high accuracy and stability. Probing the 5s(21)S(0) - 5(s)5(p)(3)P(0) transition of In-115 - at A 237 nm with a natural linewidth of 0.8 Hz by a sub-Hertz linewidth laser can lead to an accuracy of 10(-18) after several days of averaging. So far, a fractional resolution of 1.3 x 10(-13) has been obtained, limited by the frequency fluctuations of the laser exciting the (1)So - P-3(0) resonance. The absolute frequency of the S-1(0) - P-3(0) transition was determined to 1267 402 452 899.92(0.23) kHz (Deltav/v = 1.8 X 10(-13)), where the measurement uncertainty is dominated by the frequency uncertainty of the He-Ne laser serving as a reference. Further improvements in this measurement together with a comparison with other narrow atomic resonances would allow investigations of variations of fundamental constants in time.