We analyze the average performance of the least absolute shrinkage and selection operator (Lasso) for the linear model under a Gaussian matrix design, when the number of regressors grows larger while keeping the true support size finite, i.e., the ultra-sparse case. The result is based on a novel treatment of the non-rigorous replica method in statistical physics, where self-averaging assumptions on certain random variables are relaxed. Using our theory, the average performance for Gaussian measurements and noise can be assessed from the solution of a random scalar optimization problem with O(1) random elements.