'Yes, Madam, I am drunk, but in the morning I shall be sober and you will still be ugly.' Who said this classic put-down line and to whom was it said?

Winston Churchill

Sir Winston Churchill delivered the line. Several sites indicate that the unfortunate recipient was Lady Astor.

Winston Churchill also touted Margaret Thatcher. Once, Thatcher remarked, 'If you were my husband, I would put poison in your coffe.' Churchill, with his quick wit, replied, 'Madam, if you were my wife, I would drink it.'

I'm pretty sure the woman was Bessie Braddock, A Labour MP. Encarta Quotations agrees with me.

This site has a discussion on the exact wording of the quote.
http://quoteinvestigator.com/2011/08/17/sober-tomorrow/

Lady Astor also famously let her wit get ahead of her wits when she called the Eighth Army the D Day Dodgers. Having missed the terrors of Normandy because instead they fought their way across North Africa and up the length of Italy, they were unimpressed. She was immortalised in a song that probably isn't suitable for AFT (google for the lyrics if you aren't offended by soldiers' vocabulary), but is an equally powerful put down in its own way.

The famous 'poison' quote was between Churchill and Nancy, Viscountess Astor. It took place at Blenheim Palace in 1912. (Quoted in E.Langthorne, 'Nancy Astor and Her Friends').

Penguin Dictionary of Modern Quotations.

This site suggests it was indeed Bessie Braddock whom he was referring to.

http://www.telegraph.co.uk/news/politics/conservative/11155416/Sir-Winston-Churchill-the-famous-lines-that-he-never-said.html