In this paper, we provide a convergence analysis of a projection semi-implicit scheme for the simulation of fluid-structure systems involving an incompressible viscous fluid. The error analysis is performed on a fully discretized linear coupled problem: a finite element approximation and a semi-implicit time-stepping strategy are respectively used for space and time discretization. The fluid is described by the Stokes equations, the structure by the classical linear elastodynamics equations and all changes of geometry are neglected. We derive an error estimate in finite time and we prove that the time discretization error for the coupling scheme is at least $\sqrt{\delta t}$. Finally, some numerical experiments that confirm the theoretical analysis are presented.