Results are presented of a search for nonflare microwave radiation from the coronae of nearby late-type dwarf stars comparable to the sun: single stars without evidence for either a large wind or circumstellar envelope. The observing program consisted of flux measurements of six stars over a 24-h period with the VLA in the C configuration at a wavelength of 6 cm with 50 MHz bandwidth. Positive detections at 6 cm were made for Chi 1 Ori (0.6 mJy) and the flare star UV Cet (1.55 mJy), and upper limits were obtained for the stars Pi 1 UMa, Xi Boo A, 70 Oph A and Epsilon Eri. It is suggested that Chi 1 Ori, and possibly UV Cet, represent the first detected members of a new class of radio sources which are driven by gyroresonance emission, i.e. cyclotron emission from nonrelativistic Maxwellian electrons.