  Let N_1 (resp.N_2) be a nest A (resp. B) be the corresponding nest algebra,
A_0 (resp. B_0) be the subalgebra of compact operators. We prove that the nests
N_1, N_2 are isomorphic if and only if the algebras A, B are weakly-* Morita
equivalent if and only if the algebras A_0, B_0 are strongly Morita equivalent.
We characterize the nest isomorphisms which implement stable isomorphism
between the corresponding nest algebras.
