We report measurements of the temperature dependence of the surface resistivity of highquality YBa2Cu3O7 crystals at 5.95 GHz. A narrow transition with 89-K onset is observed. Resistivity in the normal state is found to be Rnorm=126×10-3 Ω. We establish an upper limit for the resistivity at 77 K of 1.0×10-3 Ω and at 2 K of 1.5×10-5 Ω. The dissipation in four crystals with a total area of about 20 mm2 is measured to be nearly independent of peak microwave magnetic field over the range of 0.5 to 93 G in sharp contrast to the strong field dependence observed in bulk sintered samples.