  We study a possible dark matter candidate in the framework of a minimal
anomalous $U(1)'$ extension of the MSSM. It turns out that in a suitable
decoupling limit the St\"uckelino, the fermionic degree of freedom of the
St\"uckelberg multiplet, is the lightest supersymmetric particle (LSP). We
compute the relic density of this particle including coannihilations with the
next to lightest supersymmetric particle (NLSP) and with the next to next to
lightest supersymmetric particle (NNLSP) which are assumed almost degenerate in
mass. This assumption is needed in order to satisfy the stringent limits that
the Wilkinson Microwave Anisotropy Probe (WMAP) puts on the relic density. We
find that the WMAP constraints can be satisifed by different NLSP and NNLSP
configurations as a function of the mass gap with the LSP. These results hold
in the parameter space region where the model remains perturbative.
