We present a deterministic distributed algorithm in the LOCAL model that finds a proper $(Δ+ 1)$-edge-coloring of an $n$-vertex graph of maximum degree $Δ$ in $\mathrm{poly}(Δ, \log n)$ rounds. This is the first nontrivial distributed edge-coloring algorithm that uses only $Δ+1$ colors (matching the bound given by Vizing's theorem). Our approach is inspired by the recent proof of the measurable version of Vizing's theorem due to Grebík and Pikhurko.