Measurements of the quality factor Q approximately 8x10(9) are reported for the whispering-gallery modes (WGM's) of quartz microspheres for the wavelengths 670, 780, and 850 nm; these results correspond to finesse f approximately 2.2x10(6) . The observed independence of Q from wavelength indicates that losses for the WGM's are dominated by a mechanism other than bulk absorption in fused silica in the near infrared. Data obtained by atomic force microscopy combined with a simple model for surface scattering suggest that Q can be limited by residual surface inhomogeneities. Absorption by absorbed water can also explain why the material limit is not reached at longer wavelengths in the near infrared.