  Chirality is a distinct signature that characterizes triaxial shapes in
nuclei. We report the first observation of chirality in the nucleus $^{135}$Pr
using a high-statistics Gammasphere experiment with the
$^{123}$Sb($^{16}$O,4n)$^{135}$Pr reaction. Two chiral-partner bands with the
configuration $\pi(1h_{11/2})^1\otimes \nu(1h_{11/2})^{-2}$ have been
identified in this nucleus. Angular distribution analyses of the $\Delta I = 1$
connecting transitions between the two chiral partners have revealed a dominant
dipole character. Quasiparticle triaxial rotor model calculations are in good
agreement with the experiment. This is the first time that both signatures of
triaxiality--chirality and wobbling--have been observed in the same nucleus.
