Here we report on the first observational investigation of the Zeeman effect in the G band around 4305 Å. Our spectropolarimetric observations of sunspots with the Zürich Imaging Polarimeter at the Istituto Ricerche Solari Locarno confirm our previous theoretical prediction that the molecular Zeeman effect produces measurable circular polarization signatures in several CH lines that are not overlapped with atomic transitions. We also find both circular and linear polarization signals produced by atomic lines whose wavelengths lie in the G-band spectral region. Together, such molecular and atomic lines are potentially important for empirical investigations of solar and stellar magnetism. For instance, a comparison between observed and calculated Stokes profiles suggests that the thermodynamical and/or magnetic properties of the photospheric regions of sunspot umbrae are horizontally structured with a component that might be associated with umbral dots.