PDS 365 is a newly detected, rapidly rotating (v sin i = 20 km s ^-1), single, low-mass giant star that with HD 233517 and HD 219025 forms a remarkable ensemble of single K giants with the unique properties of rapid rotation, very strong Li lines, an asymmetrical Hα profile, and a large far-infrared excess. Their v sin i values are between 18 and 23 km s ^-1, and their LTE Li abundances, log ε(Li), are between 2.9 and 3.9. Detailed analysis of PDS 365 reveals it to be a ∼1 M_⊙ giant with a value of ¹²C/¹³C approximately equal to 12. A clear relation between high rotational velocities and very high Li abundances for K giant stars is found only when asymmetrical Ha profiles and large far-infrared excesses are present. If we consider single K giants, we find that among rapid (v sin i ≥ 8 km s^-1) rotators, a very large proportion (∼50%) are Li-rich giants. This proportion is in contrast with a very low proportion (∼2%) of Li-rich stars among the much more common slowly rotating K giants. This striking difference is discussed in terms of proposed mechanisms for Li enrichment.