In this paper we determine the stretch factor of L1-Delaunay and L∞-Delaunay triangulations, and we show that it is equal to √(4 + 2√2) ≈ 2.61. Between any two points x, y of such triangulations, we construct a path whose length is no more than √(4 + 2√2) times the Euclidean distance between x and y, and this bound is the best possible. This definitively improves the 25-year old bound of triangulations of √10 by Chew (SoCG '86). This is the first time the stretch factor of the Lp-Delaunay for any real p ≥ 1, is determined exactly.