We have developed spherically symmetric dynamical models of dwarf spheroidal (dSph) galaxies using Schwarzschild's orbit superposition method. This type of modelling yields constraints both on the total mass distribution (e.g. enclosed mass and scale radius) and on the orbital structure of the system (e.g. velocity anisotropy). This method is thus less prone to biases introduced by assumptions in comparison to the more commonly used Jeans modelling, and it allows us to put reliable constraints on their dark matter content. Here we present our results for the Sculptor dSph galaxy, after testing our methods on mock data sets. We fit both the second and fourth velocity moment profiles to break the mass-anisotropy degeneracy. For an Navarro, Frenk & White (NFW) dark matter halo profile, we find that the mass of Sculptor within 1 kpc is M1 kpc = (1.03 ± 0.07) × 108 M⊙, and that its velocity anisotropy profile is tangentially biased and nearly constant for radii beyond ˜100 pc. The preferred concentration (c ˜ 15) is low for its dark matter mass but consistent within the scatter found in N-body cosmological simulations. When we let the value of the central logarithmic slope α vary, we find that the best-fitting model has α = 0, although an NFW cusp or shallower is consistent at the 1σ confidence level. On the other hand, very cuspy density profiles with logarithmic central slopes α < -1.5 are strongly disfavoured for Sculptor.