The in-plane resistivity rho and thermal conductivity kappa of single crystal Na0.7CoO2 were measured down to 40 mK. Verification of the Wiedemann-Franz law, kappa/T=L-0/rho as T-->0, and observation of a T-2 dependence of rho at low temperature establish the existence of a well-defined Fermi-liquid state. The measured value of coefficient A reveals enormous electron-electron scattering, characterized by the largest Kadowaki-Woods ratio A/gamma(2) encountered in any material. The rapid suppression of A with magnetic field suggests a possible proximity to a magnetic quantum critical point. We also speculate on the possible role of magnetic frustration and proximity to a Mott insulator.