Let (R, P) be a Noetherian unique factorization domain (UFD) and M be a finitely generated R -module. Let I (M) be the first nonzero Fitting ideal of M and the order of M, denoted ordR (M), be the largest integern such that I (M) ÍPn. In this paper, we show that if M is a module of order one, then either M is isomorphic with direct sum of a free module and a cyclic module or M is isomorphic with a special module represented in the text. We also assert some properties of M while ordR(M)=2.