  Using SAGBI basis techniques, we find Gr\"obner bases for the presentation
ideals of the Rees algebras and special fiber rings of unit interval
determinantal facet ideals. In particular, we show that unit interval
determinantal facet ideals are of fiber type and that their special fiber rings
are Koszul. Moreover, their Rees algebras and special fiber rings are normal
Cohen-Macaulay domains and have rational singularities.
