  We present a measurement of the total {\it WW} and {\it WZ} production cross
sections in $p\bar{p}$ collision at $\sqrt{s}=1.96$ TeV, in a final state
consistent with leptonic $W$ boson decay and jets originating from heavy-flavor
quarks from either a $W$ or a $Z$ boson decay. This analysis uses the full data
set collected with the CDF II detector during Run II of the Tevatron collider,
corresponding to an integrated luminosity of 9.4 fb$^{-1}$. An analysis of the
dijet mass spectrum provides $3.7\sigma$ evidence of the summed production
processes of either {\it WW} or {\it WZ} bosons with a measured total cross
section of $\sigma_{WW+WZ} = 13.7\pm 3.9$~pb. Independent measurements of the
{\it WW} and {\it WZ} production cross sections are allowed by the different
heavy-flavor decay-patterns of the $W$ and $Z$ bosons and by the analysis of
secondary-decay vertices reconstructed within heavy-flavor jets. The
productions of {\it WW} and of {\it WZ} dibosons are independently seen with
significances of $2.9\sigma$ and $2.1\sigma$, respectively, with total cross
sections of $\sigma_{WW}= 9.4\pm 4.2$~pb and
$\sigma_{WZ}=3.7^{+2.5}_{-2.2}$~pb. The measurements are consistent with
standard-model predictions.
