The Weyl algebra over a field
k
k
of characteristic
0
0
is a simple ring of Gelfand-Kirillov dimension 2, which has a grading by the group of integers. We classify all
Z
\mathbb {Z}
-graded simple rings of GK-dimension 2 and show that they are graded Morita equivalent to generalized Weyl algebras as defined by Bavula. More generally, we study
Z
\mathbb {Z}
-graded simple rings
A
A
of any dimension which have a graded quotient ring of the form
K
[
t
,
t
−
1
;
σ
]
K[t, t^{-1}; \sigma ]
for a field
K
K
. Under some further hypotheses, we classify all such
A
A
in terms of a new construction of simple rings which we introduce in this paper. In the important special case that
GKdim
A
=
t
r
.
d
e
g
(
K
/
k
)
+
1
\operatorname {GKdim} A = \operatorname {tr.deg}(K/k) + 1
, we show that
K
K
and
σ
\sigma
must be of a very special form. The new simple rings we define should warrant further study from the perspective of noncommutative geometry.