  For Einstein's General Relativity (GR) or the alternatives suggested up to
date the vacuum energy gravitates. We present a model where a new measure is
introduced for integration of the total action in the D-dimensional space-time.
This measure is built from D scalar fields $\varphi_{a}$. As a consequence of
such a choice of the measure, the matter lagrangian $L_{m}$ can be changed by
adding a constant while no gravitational effects, like a cosmological term, are
induced. Such Non-Gravitating Vacuum Energy (NGVE) theory has an infinite
dimensional symmetry group which contains volume-preserving diffeomorphisms in
the internal space of scalar fields $\varphi_{a}$. Other symmetries contained
in this symmetry group, suggest a deep connection of this theory with theories
of extended objects. In general {\em the theory is different from GR} although
for certain choices of $L_{m}$, which are related to the existence of an
additional symmetry, solutions of GR are solutions of the model. This is
achieved in four dimensions if $L_{m}$ is due to fundamental bosonic and
fermionic strings. Other types of matter where this feature of the theory is
realized, are for example: scalars without potential or subjected to nonlinear
constraints, massless fermions and point particles. The point particle plays a
special role, since it is a good phenomenological description of matter at
large distances. de Sitter space is realized in an unconventional way, where
the de Sitter metric holds, but such de Sitter space is supported by the
existence of a variable scalar field which in practice destroys the maximal
symmetry. The only space - time where maximal symmetry is not broken, in a
dynamical sense, is Minkowski space. The theory has non trivial dynamics in 1+1
dimensions, unlike GR.
