We study a mean-reverting model for interest rates. The model is an extension of the Vasicek model and is a sum of non-Gaussian Ornstein-Uhlenbeck processes with subordinators, i.e. Lévy processes with only positive jumps, giving variation of the interest rate. The model have the advantage that it gives only positive interest rates, contrary to the Vasicek model. We calculate explicit results for the characteristic function and the autocorrelation function of the interest rate for both general subordinators and the case where the subordinators are compound Poisson. We also find prices of zero-coupon bonds and European options written on these bonds by applying Fourier methods. It seems that the model is simple enough to allow for analytical pricing of bonds and options in addition to capture the characteristics of the interest rate. In the end we demonstrate in a simulation how the model behave with certain values of the variables.