Fourier transform spectra of near-infrared laser-induced fluorescence in (39)K(6)Li show transitions to high vibrational levels of both the X (1)Sigma(+) and a (3)Sigma(+) electronic states. These include 147 transitions into six vibrational levels of the a (3)Sigma(+) state, which lie between 7 and 88 cm(-1) below the dissociation asymptote. Unfortunately, their energies span less than 30% of the well depth. However, fitting those data to eigenvalues of analytical model potential functions whose outer limbs incorporate the theoretically predicted long-range form, V(R) approximately D-C(6)R(6)-C(8)R(8), yields complete, plausible potential curves for this state. The best fits converge to remarkably similar solutions which indicate that D(e)=287(+/-4) cm(-1) and R(e)=4.99(+/-0.09) A for the a (3)Sigma(+) state of KLi, with omega(e)=47.3(+/-1.4) and 44.2(+/-1.5) cm(-1) for (39)K(6)Li and (39)K(7)Li, respectively. Properties of the resulting potential are similar to those of a published ab initio potential and are consistent with those of the analogous states of Li(2), K(2), Na(2), and NaK.