Letf(x) be the restriction to the real axis of an entire function of exponential typeτ<π and of power growth on the axis. Then thenth order cardinal spline,ℒnf(x), interpolatingf(x) at the integers converges uniformly on compacta tof(x). This is also true of the respective derivatives. An example shows that exponential typeπ is not necessarily permitted. The proof utilizes distribution theory and estimates on the derivatives of the Fourier transform of the fundamental splineLn(x).