High-resolution cold-target recoil-ion momentum spectroscopy (COLTRIMS) has been used to study single-electron capture processes in collisions of Ar⁶⁺ ions with He at an impact energy of 9 keV and scattering angles between 0 and 3 mrad. The energy-gain spectra show that the capture occurs mainly into the 4s state with a significant contribution involving capture into the 4p state. Reaction channels associated with transfer excitation into the 3s3p3d states of Ar⁵⁺ are also observed for the first time, accounting for 6(±0.4)% of the total cross sections for single-electron capture. Experimental results are found to be in good agreement with the two-electron atomic orbital close-coupling calculations.