Yes, you do have to pay. Sigh.

ba.com will split out the fare from the charges so you could go online and price it up?

Been ages since I last went to the US so can't recall what the approximate charges are. I'm sure others will know.

Edited to add
Just had a look at some random dates in January for LHR-JFK. Charges in J and F are in the £330 range. For LHR-SFO its in the £410 range.