The article deals with the problem of estimating the parameters of a tapered Pareto distribution. Using the moment method, we obtain new estimates depending on an additional parameter. We prove that the joint asymptotic distribution of these estimates is Gaussian. A procedure is proposed that permits one to choose the additional parameter in an optimal way. The new estimates are compared with the corresponding maximum likelihood estimates. By way of example, an application of the new estimates to the COVID-19 incidence data is given. A new algorithm for a random variable generator with a tapered Pareto distribution is proposed.