Let R be a semiprime ring. Then the extended centroid of R, C(R), can be viewed as the center of the symmetric ring of quotients of R, denoted by S(R). If A is a \(C^*\)-algebra then the involution of A extends in a natural way to a positive definite involution on S(A) and, consequently, a partial order is defined in S(A) by taking as a positive cone the set of elements of the form \(\sum^{n}_{i=1}x^*_ ix_ i\), where \(x_ i\in S(A)\). It is described the bounded subring of S(A) with respect to this partial order in terms of the algebras of multipliers of the closed essential ideals of A. The author uses this description, as well as the Dauns-Hofmann Theorem to obtain a characterization of C(A) in terms of the primitive spectrum of A. By using this characterization it is proved that the extended centroid of a prime \(C^*\)-algebra is trivial and that, if the primitive spectrum of a \(C^*\)-algebra is Hausdorff, then its extended centroid is in fact the maximal quotient ring of the center of the algebra of multipliers.