We have derived beryllium abundances for 7 Li-rich giant (A(Li) > 1.5) stars and 10 other Li-normal giants, with the aim of investigating 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 studied in this paper were found to have detectable 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 Li and Be abundance data. The present results favor the idea that Li has been produced in the interior of the stars by a Cameron-Fowler process and brought up to the surface by an extra mixing mechanism.