Gröbner and Gröbner-Shirshov bases theories are generating increasing interest because of its usefulness in providing computational tools and in giving algebraic structures which are applicable to a wide range of problems in mathematics, science, engineering, and computer science. In particular, Gröbner and Gröbner-Shirshov bases theories are powerful tools to deal with the normal form, word problem, embedding problem, extensions of algebras, Hilbert series, etc. The true significance of Gröbner-Shirshov bases is the fact that they can be computed. Gröbner-Shirshov basis and normal form of the elements were already found for the Coxeter groups of type An;Bn and Dn in [1]. They also proposed a conjecture for the general form of Gröbner-Shirshov bases for all Coxeter groups. In [2], the example was given to show that the conjecture is not true in general. The Gröbner-Shirshov bases of the other nite Coxeter groups are given in [3] and [4]. This paper is the first example of finding Gröbner-Shirshov bases for an innite Coxeter group, dened by generators and dening relations. The main purpose of this paper is to find a Gröbner-Shirshov basis and as an application classify all reduced words for the Weyl group e An. The strategy for solving the problem is as follows: Even though Gröbner bases algorithms implemented in Computer Algebra systems, there is no good Computer Algebra package to compute Gröbner-Shirshov bases. Because of noncommutative structure, it is not easy to find Gröbner-Shirshov bases. We wrote a program in Mathematica to find Gröbner-Shirshov basis of e An for small n's. Then we generalize this set to any positive integer n, called it R0. After that using the algorithm of elimination of leading words with respect to the polynomials in R0, all the words in the group e An are reduced to the explicit classes of words for small n's with help of Mathematica. As before, we also generalize this reduced set to any positive integer n. Then using combinatorial techniques, we compute the number of all reduced words with respect to these classes by means of a generating function. This generating function turns out to be same with the well known Poincare polynomial of the Weyl group e An. Therefore, by the Composition-Diamond Lemma the functions in R0 form a Gröbner-Shirshov basis for the Weyl group e An. Furthermore, one can easily see that this basis is in fact a reduced Gröbner-Shirshov basis.