What is the prince's ultimatum in Shakespeare's Romeo and Juliet?

The prince’s ultimatum is that anyone who fights in the streets will be put to death. 

An ultimatum is an order that forces someone to do something.  After the huge brawl started by Sampson and Abraham in the beginning of the play, Prince Escalus decides that he is sick of fighting in the streets.  He tells Lords Capulet and Montague that enough is enough.  Their feud has shed enough blood. 

Three civil brawls, bred of an airy word,
By thee, old Capulet, and Montague,
Have thrice disturb'd the quiet of our streets,
And made Verona's ancient citizens
Cast by their grave beseeming ornaments,
To wield old partisans, in hands as old,
Canker'd with peace, to part your canker'd hate:
If ever you disturb our streets again,
Your lives shall pay the forfeit of the peace. (Act 1, Scene 1) 

When he says that the lives of anyone who is fighting will pay the forfeit of the peace, he is basically saying that if anyone is caught fighting and lives, he will be put to death.  He hopes that this sentence will cause the feuding family members to think twice before picking up swords against each other. 

It doesn’t work.  Tybalt still tries to fight Romeo, and Mercutio doesn’t care about the ban either when he fights in his place.  Romeo didn’t want to fight, but when Tybalt killed Mercutio, he stepped in.  Romeo wasn’t put to death because of his fight, but he was banished.  Romeo was not pleased with the leniency. 

ROMEO

What less than dooms-day is the prince's doom?

FRIAR LAURENCE

A gentler judgment vanish'd from his lips,
Not body's death, but body's banishment.

ROMEO

Ha, banishment! be merciful, say 'death;'
For exile hath more terror in his look,
Much more than death: do not say 'banishment.' (Act 3, Scene 3)

For Romeo, any time away from his new bride Juliet was too long.  He was very upset by the banishment even though the prince was trying to be “gentle.”  In fact, Romeo did probably deserve a lesser sentence due to the extenuating circumstances of the duel, since he was forced into fighting in self-defense, but he was still very upset about being banished.