Fluctuations in phase across the weak links in a granular superconductor contribute to the excess conductivity below Tm, the temperature at which the plot of temperature derivative of electrical resistivity against temperature shows a maximum. In a narrow range of temperature above T∞ (the temperature at which the resistivity at which the resistivity becomes zero), the excess conductivity, Δσ = [σ(T) - σ(Tm)], is expected to vary as (T-T∞)-γ. In this work, resistivity measurements on CaREBaCu3O7-y (RE=La, Sm), Ag coated YBa2Cu3O7-y, Tl2CaBa2Cu2O8 and YBa2Cu3-xZnxCu3O7-y (x=0.05, 0.075, 0.1 and 0.15) in the paracoherence region are reported. Analysis of the data indicates the excess conductivity in this region is better described by the critical exponent y = 1.33. From an analogy between the tunneling Hamiltonian in the granular material and the Hamiltonian for an XY ferromagnet one expects the critical exponent, γ, to be the same as the critical exponent for susceptibility in the ferromangetic case. © 1992.