  In models with universal extra dimensions (i.e. in which all Standard Model
fields, including fermions, propagate into compact extra dimensions) momentum
conservation in the extra dimensions leads to the conservation of Kaluza--Klein
(KK) number at each vertex. KK number is violated by loop effects because of
the orbifold imposed to reproduce the chiral Standard Model with zero modes,
however, a KK parity remains at any order in perturbation theory which leads to
the existence of a stable lightest KK particle (LKP). In addition, the
degeneracy in the KK spectrum is lifted by radiative corrections so that all
other KK particles eventually decay into the LKP. We investigate cases where
the Standard Model lives in five or six dimensions with compactification radius
of TeV$^{-1}$ size and the LKP is the first massive state in the KK tower of
either the photon or the neutrino. We derive the relic density of the LKP under
a variety of assumptions about the spectrum of first tier KK modes. We find
that both the KK photon and the KK neutrino, with masses at the TeV scale, may
have appropriate annihilation cross sections to account for the dark matter,
$\Omega_M \sim 0.3$.
