In this article we show how the universe of BST, bounded set theory can be enlarged by definable subclasses of sets so that Separation and Replacement are true in the enlargement for all formulas, including those in which the standardness predicate may occur. Thus BST is strong enough to incorporate external sets in the internal universe in a way sufficient to develop topics in nonstandard analysis inaccessible in the framework of a purely internal approach, such as Loeb measures.