Mine is eligible too. Apple sent the diagnostics test thing to an email address on the phone and they then confirmed its the battery.
They told me there is still 45days of warranty left on the phone.
My closest Apple garage is about 40 miles away, so its not so handy getting there.
One option was to pay about £480 inc £29 ish for carriage by credit card, they would deliver a new iphone5, take the old one back, examine it for water damage or whatever, take whatever is needed to repair it and refund the rest back to my card.
If there is nothing wrong with the phone other that the faulty battery the money, minus the carriage cost will be refunded back to the card.
Not so sure I like that option.