This paper investigates the completion of the maximal Op*-algebra L+ (D) of (possibly) unbounded operators on a dense domain D in a Hilbert space. It is assumed that D is a Frechet space with respect to the graph topology. Let D+ denote the strong dual of D, equipped with the complex conjugate linear structure. It is shown that the completion of L+ (D) (endowed with the uniform topology) is the space of continuous linear operators ??? (D, <7>D+) . This space is studied as an ordered locally convex space with an involution and a partially defined multiplication. A characterization of bounded subsets of D in terms of self-adjoint operators is given. The existence of special factorizations for several kinds of operators is proved. It is shown that the bounded operators are uniformly dense in L+ (D).