Sorry, but you do NOT have a 'clean record' in the eyes of the US authorities. The USA does not recognise the UK's Rehabilitation of Offenders Act and no offence ever becomes 'spent' under US law.
Theft is classed in US law as an 'offence of moral turpitude'. The US Immigration and Nationality Act states that anyone who has EVER been convicted of such an offence is permanently ineligible to enter the USA. So, in the first instance, the US Embassy must refuse you a visa. (They have no discretion. It's the law).
However US law also permits you to apply for a 'waiver of permanent ineligibility' to enter their country. So the US Embassy has firstly complied with their legal obligation not to grant you a visa but then, on your behalf, forwarded your visa application (as an application for a waiver of ineligibility) to Washington.
If the authorities in Washington grant the waiver, the US Embassy can then reconsider your application for a visa (as they'll no longer automatically be obliged to refuse the application).
I'm surprised that the letter you've received quotes a wait of around one month, as many people seem to have to wait far longer. (The longest wait I've seen reported here on AB was for a guy who had to wait 15 months before finding out that no waiver of ineligibility would be issued, so he couldn't get a visa. He had two convictions for driving without insurance).
Chris
PS: You ask "which Department". The answer to that is the Department of Homeland Security, in Washington.