Inelastic neutron scattering measurements were performed to study spin dynamics in the noncentrosymmetric antiferromagnet α-Cu_{2}V_{2}O_{7}. For the first time, nonreciprocal magnons were experimentally measured in an antiferromagnet. These nonreciprocal magnons are caused by the incompatibility between anisotropic exchange and antisymmetric Dzyaloshinskii-Moriya interactions, which arise from broken symmetry, resulting in a collinear ordered state but helical spin dynamics. The nonreciprocity introduces the difference in the phase velocity of the counterrotating modes, causing the opposite spontaneous magnonic Faraday rotation of the left- and right-propagating spin waves. The breaking of spatial inversion and time reversal symmetry is revealed as a magnetic-field-induced asymmetric energy shift, which provides a test for the detailed balance relation.