The well known Dauns-Hofmann theorem identifies the center of the multiplier algebra M(A) of a \(C^*\)-algebra A with the algebra C(\(\beta\) Ǎ) of all continuous functions on the Stone-Čech compactification \(\beta\) Ǎ of the primitive spectrum Ǎ of A. A local version of M(A) was first studied by \textit{G. A. Elliott} [J. Funct. Anal. 23, 1-10 (1976; Zbl 0335.46037)] and \textit{G. K. Pedersen} [Invent. Math. 45, 299- 305 (1978; Zbl 0376.46040)], and more recently by the authors [Arch. Math. 54, No.4, 358-364 (1990; Zbl 0669.46026); Glasgow Math. J., in press (1990); J. Austral. Math. Soc., to appear]. This local multiplier algebra \(M_{loc}(A)\) is defined as the direct limit \(\lim_{\to}M(I)\) where I runs through the downwards directed set of all closed essential ideals of A. The main result of the present paper identifies the center \(Z(M_{loc}(A))\) of \(M_{loc}(A)\) with C(\(\lim_{\leftarrow}\beta \check I)\) where the inverse limit is taken over all dense open subsets Ǐ of Ǎ. This is then applied to study the process of iterating the local multiplier algebra; in particular, it is proved that \(Z(M_{loc}(M_{loc}(A)))=Z(M_{loc}(A))\).