This paper develops an algorithm based on the Modular Approach to solve singly constrained separable discrete optimization problems (Nonlinear Knapsack Problems). The Modular Approach uses fathoming and integration techniques repeatedly. The fathoming reduces the decision space of variables. The integration reduces the number of variables in the problem by combining several variables into one variable. Computational experiments for "hard" test problems with up to 1000 variables are provided. Each variable has up to 1000 integer values.