We introduce Envelopt, a globally convergent iterative framework for a broad class of structured optimization problems where a smooth objective is augmented by a nonsmooth convex regularizer and the variables are subject to general smooth constraints. The method is akin to an augmented-Lagrangian method in which partial minimization with respect to a lifting variable results in smooth subproblems involving the Moreau envelope of the nonsmooth regularizer, and in which the original constraints are retained explicitly. Subproblems may be solved with off-the-shelf nonlinear optimization solvers. The key insight is that partial minimization over an auxiliary variable replaces the intractable nonsmooth term with its Moreau envelope, which is continuously differentiable and requires only the proximal operator of the regularizer. We state global convergence properties, establish that feasible limit points are asymptotically stationary, and develop an infeasibility detection mechanism. We derive worst-case iteration complexity bounds when the penalty parameter is and is not bounded away from zero. The framework is general enough to subsume the classical augmented Lagrangian method and naturally accommodates important extensions, including stabilized formulations for degenerate problems, exact penalty methods, and conic constraints. We provide a Julia implementation, Envelopt.jl, as part of the JuliaSmoothOptimizers ecosystem. Numerical experiments with low-rank matrix completion, semidefinite programming, complementarity-constrained optimization, and nonconvex regularizers demonstrate the effectiveness and versatility of Envelopt.