  In this paper, we study the two-parameter quantum group $U_{r,s}(\mathfrak
sl_{\infty})$ associated to the Lie algebra $\mathfrak sl_{\infty}$ of infinite
rank. We shall prove that the two-parameter quantum group $U_{r,s}(\mathfrak
sl_{\infty})$ admits both a Hopf algebra structure and a triangular
decomposition. In particular, it can be realized as the Drinfeld double of it's
certain Hopf subalgebras. We will also study a two-parameter twisted
Ringel-Hall algebra $H_{r,s}(A_{\infty})$ associated to the category of all
finite dimensional representations of the infinite linear quiver $A_{\infty}$.
In particular, we will establish an iterated skew polynomial presentation of
$H_{r,s}(A_{\infty})$ and prove that $H_{r,s}(A_{\infty})$ is a direct limit of
the directed system of the two-parameter Ringel-Hall algebras $H_{r,s}(A_{n})$
associated to the finite linear quiver $A_{n}$. As a result, we construct a PBW
basis for $H_{r,s}(A_{\infty})$ and prove that all prime ideals of
$H_{r,s}(A_{\infty})$ are completely prime. Furthermore, we will establish an
algebra isomorphism from $U_{r,s}^{+}(\mathfrak sl_{\infty})$ to
$H_{r,s}(A_{\infty})$, which enable us to obtain the corresponding results for
$U_{r,s}^{+}(\mathfrak sl_{\infty})$. Finally, via the theory of generic
extensions in the category of finite dimensional representations of
$A_{\infty}$, we shall construct several monomial bases and a bar-invariant
basis for $U^{+}_{r,s}(\mathfrak sl_{\infty})$.
