The Euler graph has vertices labelled $(n,k)$ for $n=0,1,2,\ldots$ and $k=0,1,\ldots,n$, with $k+1$ edges from $(n,k)$ to $(n+1,k)$ and $n-k+1$ edges from $(n,k)$ to $(n+1,k+1)$. The number of paths from (0,0) to $(n,k)$ is the Eulerian number $A(n,k)$, the number of permutations of $1,2,\ldots,n+1$ with exactly $n-k$ falls and k rises. We prove that the adic (Bratteli–Vershik) transformation on the space of infinite paths in this graph is ergodic with respect to the symmetric measure.