  We define the categories of weight-finite modules over the type $\mathfrak
a_1$ quantum affine algebra $\dot{\mathrm{U}}_q(\mathfrak a_1)$ and over the
type $\mathfrak a_1$ double quantum affine algebra
$\ddot{\mathrm{U}}_q(\mathfrak a_1)$ that we introduced in a previous paper. In
both cases, we classify the simple objects in those categories. In the quantum
affine case, we prove that they coincide with the simple finite-dimensional
$\dot{\mathrm{U}}_q(\mathfrak a_1)$-modules which were classified by Chari and
Pressley in terms of their highest (rational and $\ell$-dominant)
$\ell$-weights or, equivalently, by their Drinfel'd polynomials. In the double
quantum affine case, we show that simple weight-finite modules are classified
by their ($t$-dominant) highest $t$-weight spaces, a family of simple modules
over the subalgebra $\ddot{\mathrm{U}}_q^0(\mathfrak a_1)$ of
$\ddot{\mathrm{U}}_q(\mathfrak a_1)$ which is conjecturally isomorphic to a
split extension of the elliptic Hall algebra. The proof of the classification,
in the double quantum affine case, relies on the construction of a double
quantum affine analogue of the evaluation modules that appear in the quantum
affine setting.
