We establish two versions of Vizing's theorem for Borel multi-graphs whose vertex degrees and edge multiplicities are uniformly bounded by respectively $Δ$ and $π$. The ``approximate'' version states that, for any Borel probability measure on the edge set and any $ε>0$, we can properly colour all but $ε$-fraction of edges with $Δ+π$ colours in a Borel way. The ``measurable'' version, which is our main result, states that if, additionally, the measure is invariant, then there is a measurable proper edge colouring of the whole edge set with at most $Δ+π$ colours.