  While the existence of a spin-liquid ground state of the spin-1/2 kagome
Heisenberg antiferromagnet (KHAF) is well established, the discussion of the
effect of an interlayer coupling (ILC) by controlled theoretical approaches is
still lacking. Here we study this problem by using the coupled-cluster method
to high orders of approximation. We consider a stacked KHAF with a
perpendicular ILC $J_\perp$, where we study ferro- as well as antiferromagnetic
$J_\perp$. We find that the spin-liquid ground state (GS) persists until
relatively large strengths of the ILC. Only if the strength of the ILC exceeds
about 15\% of the intralayer coupling the spin-liquid phase gives way for $q=0$
magnetic long-range order, where the transition between both phases is
continuous and the critical strength of the ILC, $|J^c_\perp|$, is almost
independent of the sign of $J_\perp$. Thus, by contrast to the quantum GS
selection of the strictly two-dimensional KHAF at large spin $s$, the ILC leads
first to a selection of the $q=0$ GS. Only at larger $|J_\perp|$ the ILC drives
a first-order transition to the $\sqrt{3}\times\sqrt{3}$ long-range ordered GS.
As a result, the stacked spin-1/2 KHAF exhibits a rich GS phase diagram with
two continuous and two discontinuous transitions driven by the ILC.
