We performed 59Co nuclear magnetic resonance (NMR) measurements of single-crystalline U6Co. There is a small decrease in the Knight shift in the superconducting (SC) state, but this change mainly arises from the SC diamagnetic effect. The negligible change of the spin part of the Knight shift, together with the absence of the Pauli-paramagnetic effect in the SC U6Co, is understood as a consequence of the small spin susceptibility. The nuclear spin–lattice relaxation rate 1/T1 is also measured in the SC state under the magnetic field, and exhibits a tiny Hebel–Slichter peak just below the SC transition temperature and exponential behavior at lower temperatures. These behaviors are in agreement with the full-gap s-wave pairing in U6Co.