Raynaud made an example over a 1-dimensional non-normal noetherian local domain with no ample line bundle. But if $X$ is normal and locally noetherian then $A$ admits an $X$-ample line bundle, so by using the 3rd tensor power, $A$ is closed in a twisted projective space bundle over $X$. Indeed, connected components of $X$ are open and irreducible, so a polarization $A_{\eta} \rightarrow A^{\vee}_{\eta}$ at a generic point $\eta\in X$ extends (by normality) to an isogeny $f$ over the entire connected component of $\eta$, and $(1,f)^{\ast}(P_{A/X})$ is $X$-ample: see Ch. 1 of Faltings-Chai.
– user29283Apr 23 '13 at 14:11

Thank you very much!
– Timo KellerApr 23 '13 at 14:22

1

When the base in integral and geom. unibranch, abelian schemes are projective; see Raynaud "Faisceaux amples..." XI.1.4.
– Kestutis CesnaviciusApr 23 '13 at 16:04

Also thanks to you!
– Timo KellerApr 23 '13 at 16:45