  An N=1/2 supergravity in four Euclidean spacetime dimensions, coupled to both
vector- and scalar-multiplet matter, is constructed for the first time. We
begin with the standard (1,1) conformally extended supergravity in four
Euclidean dimensions, and freeze out the graviphoton field strength to an
arbitrary (fixed) self-dual field (the so-called C-deformation). Though a
consistency of such procedure with local supersymmetry is not guaranteed, we
find a simple consistent set of algebraic constraints that reduce the local
supersymmetry by 3/4 and eliminate the corresponding gravitini. The final field
theory (after the superconformal gauge-fixing) has the residual local N=(0,1/2)
or just N=1/2 supersymmetry with only one chiral gravitino as the corresponding
gauge field. Our theory is not `Lorentz'-invariant because of the non-vanishing
self-dual graviphoton vacuum expectation value, which is common to the
C-deformed N=1/2 rigidly supersymmetric field theories constructed in a
non-anticommutative superspace.
