This paper studies the behaviour of the solution \(u(t,\varepsilon)\) of the system of Volterra integral equations \[ \varepsilon u(t) = f(t) + \int_0^t A(t,s)u(s)\,ds, \quad 0 \leq t \leq T, \] as the positive parameter \(\epsilon\) tends to zero. Both \(f\) and \(A\) are continuous, and the eigenvalues of \(A(t,t)\) are supposed to be negative. For small values of \(\varepsilon\) this leads to a fast exponential decay within a small boundary layer. The additive decomposition technique (originally developed by O'Malley and Hoppensteadt) is used to obtain an asymptotic power series expansion of \(u(t,\epsilon)\) in the parameter \(\epsilon\). From this series an approximant is constructed with the property that the difference of the true solution and this approximant tends to zero uniformly on the interval \([0,T]\) as \(\epsilon \to 0\). The theory is illustrated by means of two examples. In the latter of these examples the decay is not exponential within the boundary layer.