The results on the measurement of electrical conductivity and magnetoconductivity of a GaAs double quantum well between 0.5 and 1.1 K are reported. The zero magnetic-field conductivity is well described from the point of view of contributions made by both the weak localization and electron-electron interaction. At low field and low temperature, the magnetoconductivity is dominated by the weak localization effect only. Using the weak localization method, we have determined the electron dephasing times ${\ensuremath{\tau}}_{\ensuremath{\phi}}$ and tunneling times ${\ensuremath{\tau}}_{t}$. Concerning tunneling, we concluded that ${\ensuremath{\tau}}_{t}$ presents a minimum around the balance point; concerning dephasing, we observed an anomalous dependence on temperature and conductivity (or elastic mean free path) of ${\ensuremath{\tau}}_{\ensuremath{\phi}}$. This anomalous behavior cannot be explained in terms of the prevailing concepts for the electron-electron interaction in high-mobility two-dimensional electron systems.