Let \(k\) be an algebraically closed field, and \(Q\) be a finite and connected quiver without oriented cycles. A sequence of finitely generated right \(kQ\)-modules \(\{E_1,\dots,E_k\}\) is called an exceptional sequence if \(\text{Hom}_{kQ}(E_j,E_i)=0\), and \(\text{Ext}^1_{kQ}(E_j,E_i)=0\) whenever \(j>i\). Such a sequence is called complete if furthermore \(n\) equals the number of points of \(Q\). C. M. Ringel had conjectured that, if \(Q\) is a Dynkin quiver and \(\{E_1,\dots,E_n\}\) is a complete exceptional sequence of \(kQ\)-modules, then \(\text{End}(\bigoplus^n_{i=1}E_i)\) is representation-finite. While this conjecture is now known to be false, it was shown by the first author to hold true if the quiver \(Q\) is the Dynkin diagram \(\mathbb{A}_n\) with a linear orientation [Algebra Colloq. 3, No. 1, 25-32 (1996; Zbl 0846.16009)]. The objective of the present paper is to show that this conjecture holds true in the more general case where \(Q\) is \(\mathbb{A}_n\) with an arbitrary orientation. More precisely, the authors show that the endomorphism algebras of complete exceptional sequences of \(kQ\)-modules are direct products of finitely many tilted algebras of type \(\mathbb{A}_m\), where \(m\leq n\). The proof is surprisingly simple, and uses perpendicular categories. The same result was shown by \textit{H. Meltzer} [in Algebras and modules II, CMS Conf. Proc. 24, 409-414 (1998; Zbl 0914.16008)].