We consider a nonparametric additive model of a conditional mean function in which the number of variables and additive components may be much larger than the sample size but the number of non-zero additive compo- nents is small relative to the sample size. The statistical problem is to determine which additive components are non-zero. The additive compo- nents are approximated by truncated series expansions with B-spline bases. The adaptive group LASSO is used to select non-zero components. We give conditions under which this procedure selects the non-zero components correctly with probability approaching one as the sample size increases. Fol- lowing model selection, oracle-efficient, asymptotically normal estimators of the non-zero components can be obtained by using existing methods. The results of Monte Carlo experiments show that the adaptive group LASSO procedure works well with samples of moderate size.