We present absolute-frequency measurements in ultracold 39 K samples of the transitions from the 4s 1/2 ground state to np 1/2 and np 3/2 Rydberg states. A global nonlinear regression of the np 1/2 and np 3/2 term values yields an improved wave number of 35 009.813 971 0(22)sys(3)stat cm −1 for the first ionization threshold of 39 K and the quantum defects of the np 1/2 and np 3/2 series. In addition, we report the frequencies of selected one-photon transitions n s 1/2 ← np 3/2 , n dj ← np 3/2 , n f j ← ndj and n g j ← nfj and two-photon transitions nf j ← npj determined by millimeter-wave spectroscopy, where j is the total angular momentum quantum number. By combining the results from the laser and millimeter-wave spectroscopic experiments, we obtain improved values for the quantum defects of the s 1/2 , d 3/2 , d 5/2 , fj and gj states. For the dj series, the inverted fine structure was confirmed for n ≥ 32. The fine-structure splitting of the f series is less than 100 kHz at n = 31, significantly smaller than the hydrogenic splitting, and the fine structure of the g series is regular for n ≥ 30, with a fine-structure splitting compatible with the hydrogenic prediction. From the measured quantum defects of the f and g series we derive an estimate for the static dipole α d and quadrupole αq polarizabilities of the K + ion core. Additionally, the hyperfine splitting of the 4s 1/2 ground state of 39 K was determined to be 461.719 700(5) MHz using radio-frequency spectroscopy and Ramsey-type interferometry.