  We consider a fully discrete loosely coupled scheme for incompressible
fluid-structure interaction based on the time semi-discrete splitting method
introduced in {\emph{[Burman, Durst \& Guzm\'an, arXiv:1911.06760]}}. The
splittling method uses a Robin-Robin type coupling that allows for a segregated
solution of the solid and the fluid systems, without inner iterations. For the
discretisation in space we consider piecewise affine continuous finite elements
for all the fields and ensure the inf-sup condition by using a
Brezzi-Pitk\"aranta type pressure stabilization. The interfacial fluid-stresses
are evaluated in a variationally consistent fashion, that is shown to admit an
equivalent Lagrange multiplier formulation. We prove that the method is
unconditionally stable and robust with respect to the amount of added-mass in
the system. Furthermore, we provide an error estimate that shows the error in
the natural energy norm for the system is $\mathcal O\big(\sqrt{T}(\sqrt{\Delta
t} + h)\big)$ where $T$ is the final time, $\Delta t$ the time-step length and
$h$ the space discretization parameter.
