Given a stationary time series X and another stationary time series Y (with a different power spectral density), we describe an algorithm for constructing a stationary time series Z that contains exactly the same values as X permuted in an order such that the power spectral density of Z closely resembles that of Y. We call this method spectral mimicry. We prove (under certain restrictions) that, if the univariate cumulative distribution function (CDF) of X is identical to the CDF of Y, then the power spectral density of Z equals the power spectral density of Y. We also show, for a class of examples, that when the CDFs of X and Y differ modestly, the power spectral density of Z closely approximates the power spectral density of Y. The algorithm, developed to design an experiment in microbial population dynamics, has a variety of other applications.