  We study a 3D fluid-rigid body interaction problem. The fluid flow is
governed by 3D incompressible Navier-Stokes equations, while the motion of the
rigid body is described by a system of ordinary differential equations
describing conservation of linear and angular momentum. Our aim is to prove
that any weak solution satisfying certain regularity conditions is smooth. This
is a generalization of the classical result for the $3D$ incompressible
Navier-Stokes equations, which says that a weak solution that additionally
satisfy Prodi - Serrin $L^r-L^s$ condition is smooth. We show that in the case
of fluid - rigid body the Prodi - Serrin conditions imply $W^{2,p}$ and
$W^{1,p}$ regularity for the fluid velocity and fluid pressure, respectively.
Moreover, we show that solutions are $C^{\infty}$ if additionally we assume
that the rigid body acceleration is bounded almost anywhere in time variable.
