We report the first operation of a Ra^{+} optical clock, a promising high-performance clock candidate. The clock uses a single trapped ^{226}Ra^{+} ion and operates on the 7s ^{2}S_{1/2}→6d ^{2}D_{5/2} electric quadrupole transition. By self-referencing three pairs of symmetric Zeeman transitions, we demonstrate a frequency instability of 1.1×10^{-13}/sqrt[τ], where τ is the averaging time in seconds. The total systematic uncertainty is evaluated to be Δν/ν=9×10^{-16}. Using the clock, we realize the first measurement of the ratio of the D_{5/2} state to the S_{1/2} state Landé g-factors: g_{D}/g_{S}=0.598 805 3(11). A Ra^{+} optical clock could improve limits on the time variation of the fine structure constant, α[over ˙]/α, in an optical frequency comparison. The ion also has several features that make it a suitable system for a transportable optical clock.