  We present a new measurement of the production cross section $\sigma(p\bar{p}
\rightarrow ZZ)$ at a center-of-mass energy $\sqrt{s}=1.96$ TeV, obtained from
the analysis of the four charged lepton final state $\ell^{+}\ell^{-}\ell^{'+}
\ell^{'-}$ ($\ell$, $\ell^{'}$ = $e$ or $\mu$). We observe ten candidate events
with an expected background of $0.37 \pm 0.13$ events. The measured cross
section $\sigma(p\bar{p} \to ZZ)= 1.26^{+0.47}_{-0.37} \mathrm{(stat)} \pm 0.14
\mathrm{(syst)}$ pb is in agreement with NLO QCD predictions. This result is
combined with a previous result from the $ZZ\to\ell^{+}\ell^{-}\nu\bar{\nu}$
channel resulting in a combined cross section of $\sigma(p\bar{p} \to ZZ)=
1.40^{+0.43}_{-0.37} \mathrm{(stat)} \pm 0.14 \mathrm{(syst)}$ pb.
