= Solution
Let $K=Q(A)$ be the <fraction field> of the <integral domain> $A$. The <Picard group of a ring> consists of isomorphism classes of finitely generated <projective modules> that have rank one after localization at every <prime ideal>. Its product is induced by the <tensor product of modules>, its identity is $[A]$, and the inverse of $[L]$ is $[L^\vee]$, where $L^\vee=\operatorname{Hom}_A(L,A)$. Evaluation $L\otimes_AL^\vee\to A$ is an isomorphism: after localization it is evaluation on a free rank-one module, and a map whose kernel and cokernel vanish at every prime is an isomorphism.
An <invertible fractional ideal> is a nonzero finitely generated $A$-submodule $I\subseteq K$ admitting another fractional ideal $J$ with $IJ=A$. The <Cartier divisors> of $\operatorname{Spec}A$ can be represented by these ideals. Concretely a <Cartier divisor> has local nonzero rational equations $f_i$ on a cover by distinguished open sets, with $f_i/f_j$ a regular unit on each overlap. We use the convention assigning to it $\mathcal O(-D)$, whose local fractional ideals are $f_iA_{s_i}$. Their unit ratios make these modules agree on overlaps. Thus the <Cartier divisor> group $\operatorname{Cart}(A)$ is realized by <invertible fractional ideals> with multiplication. The identity is $A$ and a <principal Cartier divisor> of $f\in K^\times$ corresponds to $fA$.
To check explicitly that local divisor data give an <invertible fractional ideal>, set $I=\bigcap_i f_iA_{s_i}\subseteq K$. Take a finite distinguished-open cover; this is possible because an affine spectrum is quasi-compact. On an overlap, $f_i/f_j\in A_{s_is_j}$. For each fixed $i$, clear the powers of $s_i$ in these finitely many expressions: there is $N_i$ with $s_i^{N_i}f_i\in f_jA_{s_j}$ for every $j$. Thus $t_i=s_i^{N_i}f_i$ belongs to $I$, and $I_{s_i}=f_iA_{s_i}$. The finite module $J=\sum_iAt_i$ has the same localizations, so $I/J$ vanishes on the cover and $I=J$. Repeating with $f_i^{-1}$ gives a finitely generated fractional ideal $I'$ with $I'_{s_i}=f_i^{-1}A_{s_i}$. Therefore $II'$ is contained in $A$ and equals $A$ after localization on the cover, whence $II'=A$. These conclusions use the elementary localization test proved below. Conversely, the next argument constructs precisely such a finite principal cover from an inverse ideal. Thus the local-divisor and ideal descriptions agree, without assuming that $A$ is Noetherian.
Here is the algebra underlying that realization and the map to the <Picard group>. If $IJ=A$, write $1=\sum_{i=1}^r u_iv_i$, with $u_i\in I$ and $v_i\in J$. On the distinguished open where $s_i=u_iv_i$ is invertible, $I_{s_i}=u_iA_{s_i}$: for any $x\in I$, the expression
$$
\frac{x}{u_i}=\frac{xv_i}{u_iv_i}
$$
is in $A_{s_i}$. These open sets cover the spectrum because $\sum s_i=1$. The generators on overlaps differ by units. Moreover the maps $\phi_i:I\to A$, $\phi_i(x)=v_ix$, satisfy $x=\sum u_i\phi_i(x)$. Consequently the composite
$$
I\xrightarrow{x\mapsto(\phi_i(x))}A^r
\xrightarrow{(a_i)\mapsto\sum a_iu_i}I
$$
is the identity. This explicitly makes $I$ a direct summand of a finite <free module>, and hence a <projective module>, with rank one locally.
Its inverse is
$$
I^{-1}=\{v\in K:vI\subseteq A\}=J.
$$
The inclusion $J\subseteq I^{-1}$ follows from $IJ=A$. For the other inclusion, $v\in I^{-1}$ gives $v=\sum(vu_i)v_i\in J$. Local multiplication is an isomorphism between free rank-one modules, so multiplication $I\otimes_AJ\to IJ$ is an isomorphism. In particular $I\mapsto[I]$ respects the group laws.
Conversely, suppose $L$ is a finitely generated <projective module> of rank one. It is a direct summand of a finite <free module>, so it is torsion-free and injects into $L\otimes_AK$. The latter is a one-dimensional $K$-space. Choose a basis to identify it with $K$, and let $I$ be the image of $L$. Its finitely many generators have a common nonzero denominator, making $I$ a fractional ideal. Every $A$-linear map $I\to A$ extends to a $K$-linear map $K\to K$, which is multiplication by some $v\in K$. Thus $I^\vee$ identifies with $I^{-1}$. Evaluation is an isomorphism locally, so its image ideal $II^{-1}$ localizes to $A_{\mathfrak p}$ at every prime. A proper ideal would be contained in a maximal ideal, contradicting that localization; hence $II^{-1}=A$. This proves that every <Picard group> class has an <invertible fractional ideal> representative, and also that locally principal fractional ideals on a finite open cover are invertible.
For completeness, these local tests do detect a zero module without a finite-generation assumption. If $0\ne x\in N$, its annihilator is proper; choose a maximal ideal containing it. Then $x/1$ is nonzero in that localization, since any denominator killing $x$ would belong to its annihilator. Applying this observation to a kernel and cokernel justifies the local isomorphism tests above.
The natural maps in the <Cartier divisor exact sequence for an integral domain> are now
$$
A^\times\hookrightarrow K^\times,\qquad
K^\times\longrightarrow\operatorname{Cart}(A),\quad f\longmapsto fA,
\qquad \operatorname{Cart}(A)\longrightarrow\operatorname{Pic}(A),\quad I\longmapsto[I].
$$
The first map is injective. The ideal $fA$ is the identity divisor $A$ precisely when $f$ and $f^{-1}$ lie in $A$, that is, precisely when $f\in A^\times$. Next $[I]=[A]$ precisely when some isomorphism $A\to I$ sends $1$ to an element $f\in K^\times$ generating $I$, so $I=fA$. Finally the construction from $L$ proves surjectivity onto the <Picard group>. We have therefore checked exactness at every term:
$$
\boxed{1\longrightarrow A^\times\longrightarrow K^\times
\longrightarrow\operatorname{Cart}(A)\longrightarrow\operatorname{Pic}(A)\longrightarrow0.}
$$
No Noetherian, normality or factoriality hypothesis is required.
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