Cu3(O2C16H23)6.1.2C6H12, containing a Cu36+ core in an equilateral triangle geometry, has been found to be a versatile model system for investigating the spin-frustration phenomenon in a triangular lattice. It affords well-resolved EPR spectra from both of the two possible (Stotal = 1/2 and 3/2) spin states of the Cu36+ core. From 295 to 100 K, the spectra consist of a triplet, but with the central line overlapped by an additional, sharp peak, which replaces the triplet at 30 K and below. The triplet was thus assigned to the excited state with Stotal = 3/2, located at 324 +/- 5 K ( approximately 225 cm-1), with the zero-field parameters D = -535 G, E = 0, g parallel = 2.209 and g perpendicular = 2.057. The singlet was attributed to the Stotal = 1/2 state, with gxx = 2.005, gyy = 2.050, gzz = 2.282, and, surprisingly, a hyperfine splitting arising from a single Cu2+ nucleus, with Azz = 157 G. The detailed magnetic measurements on a three-electron, equilateral triangular system, and the observation of symmetry lowering in the doublet ground state, should be of broad theoretical and experimental interest in molecular magnetism.