  We perform lattice calculations of the spatial 't Hooft k-string tensions in
the deconfined phase of SU(N) gauge theories for N=2,3,4,6. These equal (up to
a factor of T) the surface tensions of the domain walls between the
corresponding (Euclidean) deconfined phases. For T much larger than Tc our
results match on to the known perturbative result, which exhibits Casimir
Scaling, being proportional to k(N-k). At lower T the coupling becomes stronger
and, not surprisingly, our calculations show large deviations from the
perturbative T-dependence. Despite this we find that the behaviour proportional
to k(N-k) persists very accurately down to temperatures very close to Tc. Thus
the Casimir Scaling of the 't Hooft tension appears to be a `universal' feature
that is more general than its appearance in the low order high-T perturbative
calculation. We observe the `wetting' of these k-walls at T around Tc and the
(almost inevitable) `perfect wetting' of the k=N/2 domain wall. Our
calculations show that as T tends to Tc the magnitude of the spatial `t Hooft
string tension decreases rapidly. This suggests the existence of a (would-be)
't Hooft string condensation transition at some temperature which is close to
but below Tc. We speculate on the `dual' relationship between this and the
(would-be) confining string condensation at the Hagedorn temperature that is
close to but above Tc.
