We study the effects of finite spectral resolution on the magnetic field values retrieved through the weak-field approximation (WFA) from the cores of the Mg II h and k lines. The retrieval of the line-of-sight component of the magnetic field, B LOS, from synthetic spectra generated in a uniformly magnetized FAL-C atmosphere is accurate when restricted to the inner lobes of Stokes V. As we degrade the spectral resolution, partial redistribution effects, which more prominently affect the outer lobes of Stokes V, are brought into the line core through spectral smearing, degrading the accuracy of the WFA and resulting in an inference bias, which is more pronounced as the resolution becomes poorer. When applied to a diverse set of spectra emerging from a sunspot simulation, we find good accuracy in the retrieved B LOS when comparing it to the model value at the height where the optical depth in the line core is unity. The accuracy is preserved up to field strengths of B ~ 1500 G. Limited spectral resolution results in a small bias toward weaker retrieved fields. The WFA for the transverse component of the magnetic field is also evaluated. Reduced spectral resolution degrades the accuracy of the inferences, because spectral mixing results in the line effectively probing deeper layers of the atmosphere.