We consider the system modeling the motion of a rigid body into a viscous incompressible fluid. Such a system couples the Navier-Stokes system for the fluid with the Newton laws for the rigid body and has a free boundary due to the motion of the rigid body. We work here in the case where the fluid domain and the structure domain are confined into a bounded domain. We show in this article that a weak solution for this system such that the fluid velocity satisfies a Prodi-Serrin condition is smooth in time and space. As for the proof in the case of the standard Navier-Stokes system, here we consider a particular linearization of our system around our weak solution. The corresponding linear system written in a moving spatial domain is then studied with the help of the Prodi-Serrin condition and to show some uniqueness result that allows us to identify the solutions of the linear system and of the nonlinear system, we also study the adjoint of this system.