Wintry Weather Heading Our Way
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No best answer has yet been selected by grahamstuart. Once a best answer has been selected, it will be shown here.
For more on marking an answer as the "Best Answer", please visit our FAQ.It depends on whether the US regard drunkenness as being a 'crime of moral turpitude'. If so, merely being arrested for it (even if later found not guilty) is sufficient to debar you from the visa waiver program.
You'll need to ask the US Embassy and probably apply for a visa. As it won't be a straightforward case (they'll probably need to refer it to Washington for a decision) you'll need to allow plenty of time for this.
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Whether they will actually have access to records on this type of minor infringement in US immigration and would catch you out if you arrived and lied on your waiver application form I've no idea. I doubt it, but wouldn't want to risk it.