In semiconductor nanowires, the coexistence of wurtzite and zinc-blende phases enables the engineering of the electronic structure within a single material. This presupposes an exact knowledge of the band structure in the wurtzite phase. We demonstrate that resonant Raman scattering is a important tool to probe the electronic structure of novel materials. Exemplarily, we use this technique to elucidate the band structure of wurtzite GaAs at the Gamma point. Within the experimental uncertainty we find that the free excitons at the edge of the wurtzite and the zinc-blende band gap exhibit equal energies. For the first time we show that the conduction band minimum in wurtzite GaAs is of Gamma(7) symmetry, meaning a small effective mass. We further find evidence for a light-hole-heavy-hole splitting of 103 meV at 10 K.