We propose a new design method for a finite-dimensional model reference adaptive control of a class of non-linear distributed systems with input and output disturbances, when only input and output measurements are available. We consider the case where the relative degree of the dominant part of the plant is arbitrary, and introduce the idea of quasi-sliding mode in variable structure systems theory which can avoid the chattering problem. The proposed method assures that any accuracy of the output error is obtained in any finite time, independently of the presence of bounded disturbances, non-linearity and unmodelled dynamics and that all signals in the closed-loop system are uniformly bounded. Finally, we show the effectiveness of the proposed method by a simple numerical example.