For the numerical simulation of the circulatory system, geometrical multiscale models based on the coupling of systems of diﬀerential equations with diﬀerent spatial dimensions are becoming common practice [L. Formaggia et al., Comput. Vis. Sci., 2 (1999), pp. 75-83, A. Quarteroni and A. Veneziani, Multiscale Model. Simul., 1 (2003), pp. 173-195, L. Formaggia et al., Comput. Methods Appl. Mech. Engrg., 191 (2001), pp. 561-582]. In this paper we address the mathematical analysis of a coupled multiscale system involving a zero-dimensional (0D) model, describing the global characteristics of the circulatory system, and a one-dimensional (1D) model giving the pressure propagation along a straight vessel. We provide a local-in-time existence and uniqueness of classical solutions for this coupled problem. To this purpose we reformulate the original problem in a general abstract framework by splitting it into subproblems (the 0D system of ODEs and the 1D hyperbolic system of PDEs) ; then we use ﬁxed-point techniques. The abstract result is then applied to the original blood ﬂow case under very realistic hypotheses on the data. This work represents the 1D-0D counterpart of the 3D-0D mathematical analysis reported in [A. Quarteroni and A. Veneziani, Multiscale Model. Simul., 1 (2003), pp. 173-195].