We report on a measurement of the $D^{+}$-meson production cross section as a function of transverse momentum ($p_T$) in proton-antiproton ($pbar{p}$) collisions at 1.96 TeV center-of-mass energy, using the full data set collected by the Collider Detector at Fermilab in Tevatron Run II and corresponding to 10 fb$^{-1}$ of integrated luminosity. We use $D^{+} to K^-pi^+pi^+$ decays fully reconstructed in the central rapidity region $|y|<1$ with transverse momentum down to 1.5 GeV/$c$, a range previously unexplored in $pbar{p}$ collisions. Inelastic $pbar{p}$-scattering events are selected online using minimally-biasing requirements followed by an optimized offline selection. The $K^-pi^+pi^+$ mass distribution is used to identify the $D^+$ signal, and the $D^+$ transverse impact-parameter distribution is used to separate prompt production, occurring directly in the hard scattering process, from secondary production from $b$-hadron decays. We obtain a prompt $D^+$ signal of 2950 candidates corresponding to a total cross section $sigma(D^+, 1.5 < p_T < 14.5~mbox{GeV/}c, |y|<1) = 71.9 pm 6.8 (mbox{stat}) pm 9.3 (mbox{syst})~mu$b. While the measured cross sections are consistent with theoretical estimates in each $p_T$ bin, the shape of the observed $p_T$ spectrum is softer than the expectation from quantum chromodynamics. The results are unique in $pbar{p}$ collisions and can improve the shape and uncertainties of future predictions.
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