We report on measurements of differential cross sections dσ/dpT for prompt charm meson production in pp¯ collisions at 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−π+, D*+→D0π+, D+→K−π+π+, D+s→φπ+, and their charge conjugates. The measured cross sections are compared to theoretical calculations.