  We introduce an analogue of the $q$-Schur algebra associated to Coxeter
systems of type $\hat A_{n-1}$. We give two constructions of this algebra. The
first construction realizes the algebra as a certain endomorphism algebra
arising from an affine Hecke algebra of type $\hat A_{r-1}$, where $n \geq r$.
This generalizes the original $q$-Schur algebra as defined by Dipper and James,
and the new algebra contains the ordinary $q$-Schur algebra and the affine
Hecke algebra as subalgebras. Using this approach we can prove a double
centralizer property. The second construction realizes the affine $q$-Schur
algebra as the faithful quotient of the action of a quantum group on the tensor
power of a certain module, analogous to the construction of the ordinary
$q$-Schur algebra as a quotient of $U(\frak g \frak l_n)$.
