Dear Santa Clara Law Admissions,
My name is [insert name], ID [insert ID], and I was recently admitted for the Fall of [insert year]. The reason that I am sending out this e-mail is because Loyola Law School recently offered me a scholarship of $45,000 (to be spread over the 3 years of law school) that currently gives them priority to where I am probably going to attend. Now I'm not sure if it's too late, but I was wondering if it were possible for me to be considered for a scholarship at your school, because if not, I will probably end up withdrawing so that I can give another one of your applicants a spot at your law school.
Thank you for your time.
This is what I sent, worked like a charm. The best part was when I used the scholarship offer that Santa Clara gave me to increase my Loyola scholarship to 20k a year (though I didn't end up going there). Fun times.