Found the link for this right on the front page of ual.com, don't see any restrictions other than 24 hours or so to purchase...

To Paris in Style

Travel First Class to Paris at Business Class prices

Start the New Year off in style with a First Class trip to Paris. Travel to Paris in First Class for the price of a full-fare business ticket when you fly from 8 United cities.

Depart from Chicago, Denver, Los Angeles, Portland, Phoenix, San Diego, Seattle or San Francisco for Paris and enjoy First Class travel at Business prices.

This special offer is available through January 7, 2004, with the purchase of roundtrip travel.

May we remind you that Valentine’s Day is just around the corner? Perhaps you should shop now for your First Class Paris travel.