In all 2D theories of gravity a conservation law connects the (space-time d
ependent) mass aspect function at all times and all radii with an integral
of the matter fields. It depends on an arbitrary constant which may be inte
rpreted as determining the initial value together with the initial values f
or the matter field. We discuss this for spherically reduced Einstein gravi
ty in a diagonal metric and in a Bondi-Sachs metric using the first order f
ormulation of spherically reduced gravity, which allows easy and direct fix
ing of any type of gauge. The relation of our conserved quantity to the Arn
owitt-Deser-Misner (ADM) and Bondi mass is investigated. Further possible a
pplications (ideal fluid, black holes in higher dimensions or AdS space-tim
es, etc.) are straightforward generalizations.