Finitely generated projective modules over polynomial rings with integral coefficients There is famous Quillen-Suslin theorem which states that every finitely generated projective module over a ring of polynomials $k[x_1,...,x_n]$, where $k$ is a field, is free.
I have never carefully read a proof of this theorem, which is for example in the Lang's Algebra. Probably it is based on Quillen's original ideas.- this is not true as it was pointed out in the answer below.
Questions: Is every finitely generated projective modules over $\mathbb{Z}[x_1,...,x_n]$ free? 
If yes, then is the proof modification of the one given in Lang's Algebra?
And if yes, then how about polynomial rings over other Dedekind domains or number rings?

Let us say that a ring $A$ satisfies condition $(S)$ if every finite type projective $A$-module is free. Clearly a necessary condition for $(S)$ to hold is $K_0(A)=\mathbf{Z}$.
Example. (i) If $A$ is local Noetherian then $(S)$ holds.
(ii) If $A$ is a Dedekind domain then $K_0(A)=\mathbf{Z}\oplus\mathrm{Pic}(A)$, and thus a necessary condition for $(S)$ is $\mathrm{Pic}(A)=0$, which is the case iff $A$ is a pid.
Theorem.(Grothendieck) If $A$ is a regular ring then we have a canonical isomorphism $K_0(A)\rightarrow K_0(A[(T_i)_{i\in I}])$.
Thus a necessary condition for $(S)$ to hold for $A[T_1,\ldots,T_n]$ is $K_0(A)=\mathbf{Z}$.
Corollary. Let $A$ be a Dedekind domain. Then $(S)$ holds for $A[T_1,\ldots,T_n]$ if and only if $\mathrm{Pic}(A)=0$, i.e. iff $A$ is a pid.
(This follows from what we said previously and Quillen's theorem stating that if $A$ is a pid, then $A[T_1,\ldots,T_n]$ satisfies $(S)$.)
(Remark. Quillen's original proof is different from the one given in Lang!)
