  We study a cosmic string solution of an N=1-supersymmetric version of the
Cremmer-Scherk-Kalb-Ramond (CSKR) Lagrangian coupled to a vector superfield by
means of a topological mass term. The 2-form gauge potential is proposed to
couple non-minimally to matter, here described by a chiral scalar superfield.
The important outcome is that supersymmetry is kept exact both in the core and
in the exterior region of the string. We contemplate the bosonic configurations
and it can be checked that the solutions saturate the Bogomolnyi bound. A
glimpse on the fermionic zero modes is also given.
