  This Letter reports a search for a narrow resonant state decaying into two
$W$ bosons and a bottom-antibottom quark pair where one $W$ boson decays
leptonically and the other decays into a quark-antiquark pair. The search is
particularly sensitive to top-antitop resonant production. We use the full data
sample of proton-antiproton collisions at a center-of-mass energy of 1.96 TeV
collected by the CDF II detector at the Fermilab Tevatron, corresponding to an
integrated luminosity of 9.45 fb$^{-1}$. No evidence for resonant production is
found and upper limits on the production cross section times branching ratio
for a narrow resonant state are extracted. Within a specific benchmark model,
we exclude a $Z'$ boson with mass below 915 GeV/$c^2$ decaying into a
top-antitop pair at the 95% credibility level assuming a $Z'$ boson decay width
of $\Gamma_{Z'} = 0.012 M_{Z'}$. This is the most sensitive search for a narrow
$\ensuremath{q\bar{q}}$-initiated $\ensuremath{t\bar{t}}$ resonance in the mass
region below 750 GeV/$c^2$.
