The process comprises a cold absorption step 2 for removing acid gas from natural gas using a solvent phase at a pressure P.sub.2 which delivers a purified gas 5; a step for depressurizing and separating resultant acidic solvent phase containing hydrocarbons (lines 4, 11), at a pressure P.sub.12 which is lower than pressure P.sub.2 in a column-heat exchanger 12 which delivers a purified gaseous effluent 13 overhead and a further solvent phase at the bottom (line 17) which is enriched in hydrogen sulphide; and a solvent phase regeneration step carried out in a distillation column 18 at a pressure P.sub.18 and no higher than pressure P.sub.12. A gas 29 containing essentially hydrogen sulphide which is depleted in hydrocarbons is recovered from column 18 and is a suitable feed for a Claus plant for eliminating H.sub.2 S. The regenerated hot solvent phase 20 indirectly exchanges heat with the cold acidic solvent phase from absorber 2 or separator 6. It is then recycled to the absorbent after cooling.