The ESR of small concentration of Gd 0.03<y<0.06 substituting for Y in single crystals of Gd_yY_1−yBa₂Cu₃O_6−x has been measured. In the insulating compound, with x ~ 0.1, and the superdconducting materials with 30 K < T_c < 80 K, the measurements were performed at X-band, 9.3 GHz, and K_α-band, 36 GHz, over a large temperature range above T_c. 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 T_c 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.