Fractional Josephson vortices carry a magnetic flux Phi, which is a fraction of the magnetic flux quantum Phi(0) approximately 2.07 x 10(-15) Wb. Their properties are very different from the properties of the usual integer fluxons. In particular, fractional vortices in 0-kappa Josephson junctions are pinned and have an oscillation eigenfrequency which is expected to be within the Josephson plasma gap. Using microwave spectroscopy, we investigate the dependence of the eigenfrequency of a fractional Josephson vortex on its magnetic flux Phi and on the bias current. The experimental results are in good agreement with the theory.