We have searched for nuclear radio emission from a statistically complete sample of 40 Sc galaxies within 30 Mpc that are optically classified as star-forming objects, in order to determine whether weak active galactic nuclei might be present. Only three nuclear radio sources were detected, in NGC 864, NGC 4123, and NGC 4535. These galaxies have peak 6 cm radio powers of $10 20 W Hz À1 at arcsecond resolution, while upper limits of the nondetected galaxies typically range from 10 18:4 to 10 20 W Hz À1 . The three nuclear radio sources are all resolved and appear to have diffuse morphologies, with linear sizes of $300 pc. This strongly indicates that circumnuclear star formation has been detected in these three H ii galaxies. Comparisons with previous 20 cm Very Large Array (VLA) results for the detected galaxies show that the extended nuclear radio emission has a flat spectrum in two objects and is almost certainly generated by thermal emission from gas ionized by young stars in the centers of those galaxies. The 6 cm radio powers are comparable to predictions for thermal emission that are based on the nuclear H luminosities and imply nuclear star formation rates of 0.08-0.8 M yr À1 , while the low-resolution NRAO VLA Sky Survey implies galaxy-wide star formation rates of 0.3-1.0 M yr À1 in stars above 5 M . In a few of the undetected galaxies, the upper limits to the radio power are lower than predicted from the H luminosity, possibly because of overresolution of central star-forming regions. Although the presence of active nuclei powered by massive black holes cannot be definitively ruled out, the present results suggest that they are likely to be rare in these late-type galaxies with H ii spectra.