The ESR of small concentration of Gd 0.03<0.06 substituting for Y in single crystals of GdyY1−yBa2Cu3O6−x has been measured. In the insulating compound, with x0.1, and the superdconducting materials with 30 K < Tc < 80 K, the measurements were performed at X-band, 9.3 GHz, and Kα-band, 36 GHz, over a large temperature range above Tc. Angular dependence measurements exhibit a spectrum which is fully resolved in certain directions, but only partially resolved, because of exchange narrowing, in other directions. Comparisons between the spectra in the insulating and superconducting compounds shows similar angular dependent behavior. This seems to indicate that the origin of the exchange narrowing is the same in both compounds. Since this narrowing in the insulating compound arises from interaction with, or via, the Cu magnetic system, it is implied that there is a similar, perhaps fluctuating, system in the superconducting state. Preliminary measurements of the temperature dependence of the line widths may indicate the presence of spin pairing at about 110 K, above the actual Tc of 70 K. The crystal field parameters are D = 3B02 = 1307 MHz, B04 = 3.014 MHz and B44 = -11.43 MHz, for the semiconducting sample. The g-value is 1.989 0.005. These values change only slightly in the superconducting crystals.