  First measurements of cross sections for isolated prompt photon production in
deep inelastic ep scattering have been made using the ZEUS detector at the HERA
electron-proton collider using an integrated luminosity of 121 pb^-1. A signal
for isolated photons in the transverse energy and rapidity ranges 5 < E_T^gamma
< 10 GeV and -0.7 < eta^gamma < 0.9 was observed for virtualities of the
exchanged photon of Q^2 > 35 GeV^2. Cross sections are presented for inclusive
prompt photons and for those accompanied by a single jet in the range E_T^jet
\geq 6 GeV and -1.5 \leq eta^jet < 1.8. Calculations at order alpha^3alpha_s
describe the data reasonably well.
