We have derived beryllium abundances for all Li-rich giant stars known in the literature and visible from the southern hemisphere and 10 other Li-normal giants, aimed to investigate the origin of the Lithium in the Li-rich giants. In particular, we test the predictions of the engulfment scenario proposed by Siess & Livio (1999) where the engulfment of a brown dwarf or one or more giant planets would lead to a simultaneous enrichment of 7Li and 9Be. We show that regardless their nature, none of the stars observed in this work was found to have beryllium. Using simple dilution arguments we show that the engulfment of an external object as the sole source of Li enrichment is ruled out by the the Li and Be abundance data. The present results favor the idea that Li has been produced in the interior of the stars and brought up to the surface by a Cameron-Fowler mechanism.