In this paper, a class of piecewise linear Lyapunov functions is used to obtain the sufficient conditions for the stability of multiclass fluid networks under priority service disciplines. This extends and generalizes the existing work that is based on a linear Lyapunov function, and also complements the existing work that uses the piecewise linear Lyapunov function approach to study the global stability of multiclass queueing networks. A three-station network example is used to illustrate the quality of this sufficient condition. Both theoretical and experimental results indicate that the piecewise linear Lyapunov function approach yields better and effective sufficient conditions.