We define a category of bounded modules for the full toroidal Lie algebra, classify irreducible modules in this category, and give their explicit vertex operator realizations. Using the technique of vertex algebras, we prove that for a generic level such modules decompose into a tensor product of five factors, which are simple modules for a sub-VOA of a lattice VOA, two affine Lie algebras, a Heisenberg algebra, and a Virasoro algebra.