The absorption spectrum of the van der Waals molecule KAr has been recorded between 12870 and 13040 cm(-1) by means of high-resolution laser excitation spectroscopy in a supersonic jet expansion. About 1600 molecular absorption lines could successfully be assigned to the rovibrational structure of the electronic transition A (2) Pi<--X(2) Sigma(+) of (KAr)-K-39. In addition, a few absorption lines due to the isotopomer (KAr)-K-41 have been detected. Spectroscopic parameters have been obtained for vibrational levels 0...3 of X (2) Sigma(+) and 6...11 of A (2) Pi. Additional experimental information was provided by observing the spectral distribution of the fluorescence. The interatomic potentials of the X (2) Sigma(+) and the A (2) Pi states have been derived from the experimental data by means of a fully quantum-mechanical approach using suitable analytical functions. Our final results for equilibrium distance R(e) and well-depth D-e are 5.404(5) Angstrom and 40.1(6) cm(-1) for X (2) Sigma(+), 3.37(3) Angstrom and 405(15) cm(-1) for A (2) Pi(1/2), and 3.34(3) Angstrom and 427(15) cm(-1) for A (2) Pi(3/2). In addition, we could deduce qualitative information on the B-2 Sigma(+) interaction potential from our experimental data yielding R(e)=7.10 Angstrom and D-e=23 cm(-1) as preliminary values.