  In this paper we use Block's classification of simple modules over the first
Weyl algebra to obtain a complete classification of simple weight modules, in
particular, of Harish-Chandra modules, over the 1-spatial ageing algebra
age(1). Most of these modules have infinite dimensional weight spaces and so
far the algebra age(1) is the only Lie algebra having simple weight modules
with infinite dimensional weight spaces for which such a classification exists.
As an application we classify all simple weight modules over the
(1+1)-dimensional space-time Schrodinger algebra S that have a simple
age(1)-submodule thus constructing many new simple weight S-modules.
