We have recorded extremely weak absorption in the overtone band 5-0 of (CO)-C-12-O-16 X (1)Sigma(+) near 0.96 mu m with cavity ringdown spectroscopy; the light source was a Raman-shifted dye laser pumped with a frequency doubled Nd:YAG laser. This band shows lines in branch P to be much more intense than corresponding lines in branch R, in contrast to all lower overtone bands v-0 (v = 1-4). This reversal in relative intensity is explained quantitatively in terms of a radial function for the electric dipolar moment of CO. We have estimated absorption line strengths for P-3-P-18 in band 5-0 of (CO)-C-12-O-16; these strengths are consistent with a pure vibrational matrix element (5 vertical bar p(x)vertical bar 0) = (3.6 +/- 0.3) x 10(-36) C m of the electric dipolar moment p(x), a Herman-Wallis coefficient C-0(5) of about -0.1, and a band strength of (5.1 +/- 1.3) x 10(-29) m at 293 K.