We show by inelastic neutron scattering that dynamic spin correlations coexist with superconductivity in YBa2Cu3O6+x (x = 0.4,0.45,0.5). For the x = 0.5, T(c) = 50 K sample, the inelastic magnetic intensity at DELTA-E = 6 meV does not show any change near T(c) and is approximately constant from 250 to 10 K, in contrast to earlier data on superconducting La1.85Sr0.15CuO4 (T(c) = 35 K) which showed a decrease of the integrated intensity at a temperature well above T(c). The inelastic magnetic cross section observed here can be consistently modeled as two-dimensional spin waves overdamped by short correlation lengths. The correlation length in the x = 0.5 sample is an order of magnitude smaller than in the x = 0.40 and x = 0.45 samples, indicating a drastic disruption of planar copper spin-spin correlation upon hole doping. The present results are compared with recent NMR studies of T(c) = 60 K material.