A process for purifying argon to obtain a high-purity argon from argon gas containing at least nitrogen, carbon monoxide, oxygen, and methane, including a first step of adding air or oxygen to the argon gas in an amount sufficient to oxidize the contained carbon monoxide into carbon dioxide in the presence of a catalyst while maintaining a temperature sufficient to oxidize the contained carbon monoxide without substantially oxidizing the methane, a second step of adding hydrogen into the argon gas obtained in the first step and reacting the contained oxygen with the hydrogen into water in the presence of a catalyst, a third step of removing the carbon dioxide and the water from the argon gas obtained in the second step by use of an adsorbent, and a fourth step of cooling the argon gas obtained in the third step, introducing the argon thus cooled into a distillation column and performing distillation by use of a reflux containing argon as a main component to remove nitrogen, hydrogen, and methane, thereby recovering a high purity argon gas.