The two-dimensional directed spanning forest (DSF) introduced by Baccelli and Bordenave is a planar directed forest whose vertex set is given by a homogeneous Poisson point process N$\mathcal{N}$ on R2${\mathbb{R}}^2$. If the DSF has direction −ey$-e_y$, the ancestor h(u)$h(\mathbf{u})$ of a vertex u∈N$\mathbf{u}\in \mathcal{N}$ is the nearest Poisson point (in the L2$L_2$ distance) having strictly larger y$y$-coordinate. This construction induces complex geometrical dependencies. In this paper, we show that the collection of DSF paths, properly scaled, converges in distribution to the Brownian web (BW). This verifies a conjecture made by Baccelli and Bordenave in 2007 (Ann. Appl. Probab. 17 (2007) 305–359).The two-dimensional directed spanning forest (DSF) introduced by Baccelli and Bordenave is a planar directed forest whose vertex set is given by a homogeneous Poisson point process N$\mathcal{N}$ on R2${\mathbb{R}}^2$. If the DSF has direction −ey$-e_y$, the ancestor h(u)$h(\mathbf{u})$ of a vertex u∈N$\mathbf{u}\in \mathcal{N}$ is the nearest Poisson point (in the L2$L_2$ distance) having strictly larger y$y$-coordinate. This construction induces complex geometrical dependencies. In this paper, we show that the collection of DSF paths, properly scaled, converges in distribution to the Brownian web (BW). This verifies a conjecture made by Baccelli and Bordenave in 2007 (Ann. Appl. Probab. 17 (2007) 305–359).