The invention relates to a process for manufacturing a hollow glass article comprising at least one thin side wall (42), in which process: at least one parison of molten glass is introduced into a blank mold, a blank of the article is formed in the blank mold from said parison, the blank of the article is transferred into a finishing mold, and the article is formed in the finishing mold. During forming of the article, excess glass is directed towards the exterior of the side wall of the article to form a convex bulge (43) and the bulge is removed to give the side wall of the article a substantially constant thickness.The invention also relates to a corresponding machine and product.