Vibrational levels up to nu=107 of the (1)(IIg)-I-1 state of K-39(2), which dissociates to K(4s)S-2(1/2) +K(4p)P-2(3/2), have, been observed in high-resolution Fourier transform records of C (IIu)-I-1-->(1)(IIg)-I-1 fluorescence. From a revised analysis of this system, we have constructed a rotationless Rydberg-Klein-Rees (RKR) potential curve which extends to R=40.3 Angstrom. Vibrational and rotational constants have been calculated from this curve. The parameters describing the Coulombic interactions between K(4s)+ K(4p) atoms are given in a Hund＇s case (c) basis. From this work, the dissociation energy D-e for the (1)(IIg)-I-1 state is found to be 1290.30+/-0.02 cm(-1) leading to a new value of T-e=16 203.25+/-0.09 cm(-1) for this state.