ag.algebraic geometry - Divisors on Proj(UFD)

Well, if you read on to Chapter 2, exercise 6.3, then it is stated that:
$$Cl(A) cong Cl(X)/mathbb Z[H]$$
here $[H]$ represents the hyperplane section. So the answer is yes.

There is a less well-known but very nice generalization. Suppose that $X$ is smooth. Let $R=A_m$ be the local ring of A at the irrelevant ideal. Then one has a (graded) isomorphism of $mathbb Q$- vector spaces:

$$CH(X)_{mathbb Q}/[H]CH(X)_{mathbb Q} cong A_*(R)_{mathbb Q}$$

Here $CH(X)$ is the Chow ring of $X$ and $A_*(R)$ is the total Chow group of $R$. Details can be found in this paper by Kurano.