We show that under suitable conditions interpolation between a Banach space and its dual yields a Hilbert space at $theta =frac{1}{2}$. By application of this result to the special case of the non-commutative $L^p$-spaces of Leinert [Int. J. Math. textbf{2} (1991), no. 2, 177--182] and Terp [J. Operator Theory textbf{8} (1982), 327--360] we conclude that $L^2$ is a Hilbert space and that $L^p$ is isometrically isomorphic to the dual of $L^q$ without using the isomorphisms of these spaces to $L^p$-spaces of Hilsum [J. Funct. Anal. textbf{40} (1981), 151--169.] and Haagerup [Colloq. Internat. CNRS, 274, CNRS, Paris, 1979].\Haagerup and Pisier [Canad. J. Math. textbf{41} (1989), no. 5, 882--906.], Pisier [Mem. Amer. Math. Soc. textbf{122} (1996), no. 585, viii+103 pp] and Watbled [C. R. Acad. Sci. Paris, t. 321, S'erie I, p. 1437--1440, 1995] gave conditions under which interpolation between a Banach space and its conjugate dual yields a Hilbert space at $frac{1}{2}$. The result mentioned above when put in ``conjugate form'' extends their results.