Let X be a compact connected Riemann surface of genus g ≥ 2, and let MDHbe the rank one Deligne-Hitchin moduli space associated to X. It is known that MDHis the twistor space for the hyper-Kähler structure on the moduli space of rank one holomorphic connections on X. We investigate the group Aut(MDH) of all holomorphic automorphisms of MDH. The connected component of Aut (MDH) containing the identity automorphism is computed. There is a natural element of H2(MDH;ℤ). We also compute the subgroup of Aut(MDH) that fixes this second cohomology class. Since MDHadmits an ample rational curve, the notion of algebraic dimension extends to it by a theorem of Verbitsky. We prove that MDH is Moishezon. © 2017, Institute of Mathematics. All rights reserved.