Detailed spectrophotometry in the visual and near-infrared is presented for four different slit positions over the core of NGC 604. The excitation ([O II]/[O III]) varies considerably both along the slit and between different slit positions, but the line ratios ([O II]+[O III])/Hβ, [N II]/[O II], [N II]/[S II] and ([O II]/Hβ)×([O II]/[S II]) are effectively constant, confirming the suitability of these line ratios as abundance-related parameters characterizing an entire giant H II region. The [O III] electron temperature shows significant variations across the nebula, but no significant variations in relative abundances of O, N, Ne and S are found. By introducing a new ionization correction scheme the authors determine a helium abundance Y = 0.260±0.013 and this is also constant except possibly in one place where the correction for neutral helium is very uncertain. The spectrum is consistent with ionizing stars having a representative effective temperature of 40000K (blanketed LTE models) or 37000K (non-LTE models).