We report measurements of the P-3(2) <-- P-3(1) fine-structure transitions of both neutral atomic carbon species, C-12 and C-13, near 809 GHz with the Cologne terahertz spectrometer. The carbon atoms were produced in a DC glow discharge of a CO and He mixture at liquid nitrogen temperature. The C-13 atoms were generated from an isotopically enriched (CO)-C-13 sample. The P-3(2) <-- P-3(1) transition frequencies measured in absorption are determined to be C-12: J = 2 <-- 1: 809341.97 (5) MHz, C-13: J = 2 <-- 1, F = 3/2 <-- 1/2: 809125.5 (4) MHz, F = 5/2 <-- 3/2: 809493.7(2) MHz, F = 3/2 <-- 3/2: 809121.3 (4) MHz (calculated). These precise laboratory data are recommended for use as rest frequencies in future astronomical observations. The new value for the rest frequency of the C-12 (J = 2 <-- 1) transition removes a long-standing uncertainty and small discrepancy between existing laboratory data and astronomical measurements. The C-13 rest frequencies have supported the first interstellar detection of this species, as reported by Keene et al.