UK Citizen getting married in the states

Hi,

I'm a UK citizen and I live in the UK with my Fiance who is a US citizen on a student visa, soon to transfer to a spousal visa once we are married.

I wanted to know if I can legally get married to her in the state of Florida on a visitor visa / ESTA?

We have already paid for our wedding event, party etc... Just wondering if we will run into any problems in all of the legalities of getting married in another country.

Neither of us have any intension to live in the USA / apply for a green card, we both have strong ties in the UK, I hold a full time job and am enrolled on a 5 year - part time university programme in London and my fiancé holds a part time permanent position and is at university too.