We address the Primary User (PU) detection (spectrum sensing) problem, relevant to cognitive radio, from a finite random matrix theoretical (RMT) perspective. Utilizing recently-derived closed-form and exact expressions for the distribution of the standard condition number (SCN) of dual random Wishart matrices, we design a new blind algorithm to detect the presence of PU signals. An inherent property of the technique, which is due to the reliance on SCN's, is that no SNR estimation or any other information on the PU signal is required. Like some similar asymptotic RMT-based techniques recently proposed, the algorithm also admits for a tolerated probability of false alarm α to be accounted for by design. The proposed finite RMT-based algorithm, however, outperforms all known similar alternatives, in consequence of the fact that the distribution of SCN's utilized are in closed-form and exact, for any given matrix size.