This paper is concerned with the stability for a class of uncertain linear neutral systems. The uncertainty under consideration is of norm-bounded type. The discrete delay is assumed to be a time-varying continuous function belonging to a given interval, which means that the lower and upper bounds for the time-varying discrete delay are available, and no restriction on the derivative of the time-varying discrete delay is needed, which allows the discrete delay to be a fast time-varying function. Based on an integral inequality, which is introduced in this paper, and Lyapunov-Krasovskii functional approach, some stability criteria, which are formulated in the form of linear matrix inequalities (LMIs), are derived without using model transformation and bounding techniques on the related cross product terms. By the obtained criteria one can compute the allowed delay bound to guarantee the asymptotic stability of the considered systems. A numerical example is given to demonstrate effectiveness of the proposed criteria.