This paper is concerned with finding the global homological dimension of the ring of differential operators
R
[
θ
]
R[\theta ]
over a differential ring
R
R
with a single derivation. Examples are constructed to show that
R
[
θ
]
R[\theta ]
may have finite dimension even when
R
R
has infinite dimension. For a commutative noetherian differential algebra
R
R
over the rationals, with finite global dimension
n
n
, it is shown that the global dimension of
R
[
θ
]
R[\theta ]
is the supremum of
n
n
and one plus the projective dimensions of the modules
R
/
P
R/P
, where
P
P
ranges over all prime differential ideals of
R
R
. One application derives the global dimension of the Weyl algebra over a commutative noetherian ring
S
S
of finite global dimension, where
S
S
either is an algebra over the rationals or else has positive characteristic.