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For cognitive radios (CRs), compressive sampling (CS) techniques have been utilized for spectrum sensing in order to alleviate the high signal acquisition costs in the wideband regime. Given the desired sensing performance, the fundamental limit on the sampling rates is determined by the actual sparsity order Snz of the signal spectrum, which can be considerably lower than the Nyquist sampling rate. However, Snz is time-varying and hence unknown a priori for a dynamic CR network, and is typically available in the form of its statistical upper bound Smax. When the practical sampling rate is chosen according to Smax in lieu of Snz, this rate is unnecessarily high for accurate spectrum sensing and hence wasteful of the sensing resources. To circumvent this problem, this paper develops a two-step compressed spectrum sensing (TS-CSS) scheme for efficient wideband sensing. The first step quickly estimates the actual sparsity order of the wide spectrum of interest using a small number of samples, and the second step adjusts the total number of samples collected according to the estimated signal sparsity order. By doing so, the overall sampling rate is minimized adaptively. The cornerstone of this work is the fact that the number of measurements required for estimating the signal sparsity order is (much) smaller than that for reconstructing the sparse signal itself. The gap between these two numbers is delineated in this paper in closed form, which offers valuable design guideline. Validated by simulations, the proposed TS-CSS scheme achieves the desired sensing performance at considerably reduced sensing costs, using a lowered average sampling rate compared with traditional one-step compressive sampling schemes.