  (Abridged) We present results of XMM-Newton observations of the RS CVn binary
$\sigma^2$ Coronae Borealis. The RGS and EPIC MOS2 spectra were simultaneously
fitted with collisional ionization equilibrium plasma models to determine
coronal abundances of various elements. Contrary to the solar first ionization
potential (FIP) effect in which elements with a low FIP are overabundant in the
corona compared to the solar photosphere, and contrary to the ``inverse'' FIP
effect observed in several active RS CVn binaries, coronal abundance ratios in
$\sigma^2$ CrB show a complex pattern as supported by similar findings in the
Chandra HETGS analysis of $\sigma^2$ CrB with a different methodology (Osten et
al. 2003). Low-FIP elements ($<10$ eV) have their abundance ratios relative to
Fe consistent with the solar photospheric ratios, whereas high-FIP elements
have their abundance ratios increase with increasing FIP. We find that the
coronal Fe abundance is consistent with the stellar photospheric value,
indicating that there is no metal depletion in $\sigma^2$ CrB. However, we
obtain a higher Fe absolute abundance than in Osten et al. (2003). Except for
Ar and S, our absolute abundances are about 1.5 times larger than those
reported by Osten et al. (2003). However, a comparison of their model with our
XMM-Newton data (and vice versa) shows that both models work adequately in
general. We find, therefore, no preference for one methodology over the other
to derive coronal abundances. Despite the systematic discrepancy in absolute
abundances, our abundance ratios are very close to those obtained by Osten et
al. (2003). Finally, we confirm the measurement of a low density in \ion{O}{7}
($< 4 \times 10^{10}$ cm$^{-3}$), but could not confirm the higher densities
measured in spectral lines formed at higher temperatures.
