As Dzug2 states, you could 'chance it' and apply for an ESTA. (According to the rules on the US Embassy's website, you ceased to be eligible for visa-free entry to the USA at the moment when you were arrested. It wouldn't have even mattered if you were totally innocent). The US authorities don't have direct access to UK criminal records, so you might be able to get into the USA by simply keeping quiet about your conviction. However certain information is passed on by the UK authorities, and the US immigration service might still get to know about your conviction.
One possible route for information indirectly finding out about convictions is that UKBA officials, at your departure airport, might note that your passport is 'flagged' on their system as being held by someone convicted of possession of Class A drugs. They might then check whether you've got the necessary visa. If you've not, they might either advise the airline (who could be heavily fined by the US authorities if they permitted you to travel without the correct documentation) or the US immigration service.
If you decide to go 'by the book', and apply for a visa, you need to be aware that the US Immigration & Nationality Act automatically bars you from entering the USA. So, in the first instance, the US Embassy would have to reject your application. However it could then be referred to Washington, as an application for a 'waiver of permanent ineligibility'. The whole process takes a minimum of 5 months, with longer waits frequently being reported here on AB. (The longest wait I've seen was one of 15 months, for someone who'd been convicted twice for driving without insurance. He had to wait 15 months before hearing that his application for a visa had been refused).
Chris