This paper consists of two parts. In the first part, the authors prove a global existence theorem for weak solutions of a quasilinear strongly coupled parabolic system using a weak compactness method: the key step is an integral estimate on the difference quotient in time of the Galerkin approximating solutions. The second part of this work is devoted to proving the global existence and asymptotic behaviour of solutions of an electrochemistry model studied in Y. S. Choi and R. Lui [J. Differ. Equations 116, 306–317 (1995; Zbl 0819.35079)]. The charged ions are assumed to satisfy the electro-neutrality condition so, in one space dimension, the system can be transformed to a form suitable to apply the theory of the first part, except for the fact that the coefficient matrix is discontinuous at places where all the charged ions vanish. They circumvent this difficulty by constructing an approximating systems of equations and then passing to the limit. The non-negativity of weak solutions and \(L^2\)-stability of the steady state solutions are also shown under additional hypotheses.