We report on measurements of differential cross sections dsigma/dp(T) for prompt charm meson production in ppmacr; collisions at sqrt[s]=1.96 TeV using 5.8+/-0.3 pb(-1) of data from the CDF II detector at the Fermilab Tevatron. The data are collected with a new trigger that is sensitive to the long lifetime of hadrons containing heavy flavor. The charm meson cross sections are measured in the central rapidity region |y|</=1 in four fully reconstructed decay modes: D0-->K-pi(+), D(*+)-->D0pi(+), D+-->K-pi(+)pi(+), D(+)(s)-->phipi(+), and their charge conjugates. The measured cross sections are compared to theoretical calculations.