be as honest and as professional as possible. tell them exactly what you've told us. also express your gratitude and appreciation for having you, even if you don't entirely mean it. Also, you could tell them what you going to make at the other company and see if they'll match it. thats a longshot though.
Fivezero has it right on the money... it would be great if they gave you a counter offer for more $$!!
salary-matching is great...until the decide to let you go after one month. then what?