Altermagnets constitute a novel, third fundamental class of collinear magnetic ordered materials, alongside with ferro- and antiferromagnets. They share with conventional antiferromagnets the feature of a vanishing net magnetization. At the same time they show a spin-splitting of electronic bands, just as in ferromagnets, caused by the atomic exchange interaction. On the other hand, topology has recently revolutionized our understanding of condensed matter physics, introducing new phases of matter classified by intrinsic topological order. Here we connect the worlds of altermagnetism and topology, showing that the electronic structure of the altermagnet CrSb is topological and hosts a novel Weyl semimetallic state. Using high-resolution and spin angle-resolved photoemission spectroscopy, we observe a large momentum-dependent spin-splitting in CrSb, reaching up to 1 eV, that induces altermagnetic Weyl nodes with an associated magnetic quantum number. At the surface we observe their spin-polarized topological Fermi-arcs. This establishes that in altermagnets the large energy scale intrinsic to the spin-splitting -orders of magnitude larger than the relativistic spin-orbit coupling- creates its own realm of robust electronic topology.