Let \(R\) be a noetherian normal domain with quotient field \(K\) and perfect residue class fields for all prime ideals of height one, and \(L| K\) a field extension such that the integral closure \(S\) of \(R\) in \(L\) is finite over \(R\). For a tame \(R\)-order \(A\) in an Azumaya \(K\)-algebra \(\Sigma\), the authors define a ``blowing up'' \(B\) of \(A\) along \(S\), that is, a tame \(S\)-order \(B\) in \(\Sigma \otimes_ K L\) such that for a prime ideal \(P\) in \(S\) of height one, the ramification index of \(B\) at \(P\) is \(e/(e,e')\), where \(e\) (resp. \(e'\)) denotes the ramification index of the localization \(A_{P\cap R}\) (resp. of \(S\) at \(P\)). By means of this concept, the authors elucidate \textit{M. Artin}'s classification of two- dimensional maximal orders having formal power series rings over \(\mathbb{C}\) as centers [Invent. Math. 84, 195-222 (1986; Zbl 0591.16002)].