Paul (Kyle MacLachlan) takes Irulan (Virginia Madsen) as his wife, but in name only. This scene appears to have been completed, and includes music. It also appears on the old Fleer trading cards, and so was likely cut from the film at the last minute.

See scene 283 of the Seventh Draft of Dune.
The full 7th draft script of Dune is available from www.duneinfo.com

283. INT. GREAT HALL - ARRAKEEN PALACE - DAY

...

PAUL
Now, I will tell you how it will be.
Irulan shall be my wife, opening the way
for an Atreides to take the throne.

EMPEROR
I sit on the throne!!

PAUL
You will sit on a throne on Selusa
Secundus, your prison planet. Either that
or you will die.

The Emperor is still. Paul moves to Chani.

PAUL (CONT'D)
(to Chani)
The Princess shall have no more of me than
my name... no child of mine or touch the
softness of glance... nor instant of
desire.... This is my promise to you...
(whispering)
...my love.

Paul turns back to the others.

Jessica turns to Chani and whispers to her.

JESSICA
Think on it Chani... We who carry the name
of concubine... history will call us
wives.