A detailed understanding of the origin of the magnetism in dilute magnetic semiconductors is crucial to their development for applications. Using hard X-ray angle-resolved photoemission (HARPES) at 3.2 keV, we investigate the bulk electronic structure of the prototypical dilute magnetic semiconductor Ga_0.97Mn_0.03As, and the reference undoped GaAs. The data are compared to theory based on the coherent potential approximation and fully relativistic one-step-model photoemission calculations including matrix-element effects. Distinct differences are found between angle-resolved, as well as angle-integrated, valence spectra of Ga_0.97Mn_0.03As and GaAs, and these are in good agreement with theory. Direct observation of Mn-induced states between the GaAs valence-band maximum and the Fermi level, centred about 400 meV below this level, as well as changes throughout the full valence-level energy range, indicates that ferromagnetism in Ga_1−xMn_xAs must be considered to arise from both p–d exchange and double exchange, thus providing a more unifying picture of this controversial material.