As mentioned in the comments, the translation is shown in:
Volker Diekert and Paul Gastin. "First-order definable languages." (2008)
And it goes via a characterization of $LTL$ as $FO[<]$.
Sorry to provide the same answer as for this question,
but this was proved in [1, Theorem 3.1].
 J. Cohen, D. Perrin and J.-É. Pin, On the expressive power of temporal logic for finite words, J. Comput. System Sci. 46 (1993), 271-294.