The invention is a process for producing ethanol from sugarcane bagasse, the principal steps of which are mild sulphuric acid hydrolysis of the hemicellulose fraction of the sugarcane bagasse, followed by extraction of the hydrolysate and then fermentation thereof with the yeast Pichia stipitis. The process can be carried out with different solid : liquid ratios, and provides a step of acclimatizing the Pichia stipitis yeast, which results in a greater rate of ethanol production. The process takes place in a press reactor specially designed for this purpose, which allows more efficient extraction of the hydrolysate and, as a consequence, better process performance.