This paper is concerned with the internal stabilization of the generalized Korteweg de Vries equation on a bounded domain. The global well-posedness and the exponential stability are investigated when the exponent in the nonlinear term ranges over the interval [1, 4). The global exponential stability is obtained whatever the location where the damping is active, confirming positively a conjecture of Perla Menzala, Vasconcellos, and Zuazua [Quart. Appl. Math., 60 ( 2002), pp. 111-129].