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Altermagnetism, a novel class of magnetic order, has recently attracted the scientific interest. Combining the favourable properties of ferromagnets, such as spin polarised conduction electrons, with the advantages of antiferromagnets, such as zero net magnetisation and high-frequency switching, it is a promising area of research. While the ground state properties of altermagnets have been studied, their dynamic properties, such as magnon dispersions, remain elusive. Understanding these properties includes the investigation of magnon excitations, which is essential for the development of spintronic and magnonic devices based on altermagnets. In this study, we measure the energy splitting of altermagnetic magnons with opposite chirality in the hexagonal altermagnet CrSb using a unique combination of angle-resolved photoemission spectroscopy and resonant inelastic X-ray scattering. We selectively excite electronic and magnonic states with opposite spins and quantify their energy splitting. We find that the non-degeneracy of electronic and magnonic states exists away from high-symmetry directions in good agreement with recent ab initio calculations. Our experimental proof of chiral magnons demonstrates a new degree of freedom in magnetism that is both of fundamental importance and opens the door to the exploration of the non-reciprocal transport properties of these bosons in devices.