In this paper, we consider the interaction between a rigid body and an incompressible, homogeneous, viscous fluid. This fluid-solid system is assumed to fill the whole space $\Bbb R^d$, $d=2$ or $3$. The equations for the fluid are the classical Navier-Stokes equations whereas the motion of the rigid body is governed by the standard conservation laws of linear and angular momentum. The time variation of the fluid domain (due to the motion of the rigid body) is not known {\it a priori}, so we deal with a free boundary value problem. \endgraf We improve the known results by proving a complete wellposedness result: our main result yields a local in time existence and uniqueness of strong solutions for $d=2$ or $3$. Moreover, we prove that the solution is global in time for $d=2$ and also for $d=3$ if the data are small enough.