Actually, there are three options.
1. The FAA implicitly renders your UK-issued EASA PPL valid for flight on an N-reg, but only in UK airspace. You don't need to do anything for this as it's an implicit validation. But the validation is only valid within the airspace of the country that issued the license. The US doesn't recognize EASA-land as one country, and the wording is "issued" not "issued or rendered valid". So this means you can't take the N-reg abroad, not even to another EASA country.
http://www.pprune.org/private-flying...ence-reqs.html
2. You can get a "piggyback" license, which is "based on" your current EASA license. This is a relatively straightforward procedure which can easily be accomplished in the US, and with a bit more involvement in the UK. This is great for an occasional holiday flying, but for a permanent solution you need to consider that you're building a house of cards: Your US PPL will only be valid as long as your EASA PPL is valid. So there's more bits of paper to worry about, and keep valid.
Redirecting
Note that the FAA will contact the CAA in order to validate your details, but the UK CAA is not allowed, by default, to divulge your data to the FAA due to privacy legislation. So you need to send some paperwork to the UK CAA first, to allow them to divulge your data to the FAA. Read the instructions carefully - if you do things the wrong way round it will not work, or take a massive amount of time.
3. You can get a "standalone" license, which requires the FAA PPL written exam plus the flight test. Any training done for an ICAO PPL will count towards the training requirements, but it's likely you'll need some additional training since the FAA PPL includes a few maneuvers that are not included in the EASA PPL syllabus. Furthermore, the FAA PPL includes privileges to fly at night by default - there is no separate night rating for a US PPL. So there are specific experience requirements wrt. night flying as well.
In addition to this you will also need to consider the type of medical. Depending on the option you choose, you may need an FAA class 3 in addition to the EASA class 2. This can typically sorted (for a small additional fee) by your friendly AME. And depending on the option you also may need to do a BFR every two years or otherwise you're not allowed to exercise the privileges of the FAA PPL.
There's loads of information about all this on PPRuNe. The link I gave you is a good starting point.