It looks like the news has changed regarding Hearthstone's Classic mode from when Dean Ayala talked about it last week! Dean had previously stated the new mode would be going live alongside the release of the Forged in the Barrens and not patch 20.0 which was our initial guess. An update to Blizzard's core set blog since then has stated this is not the case!

This is going to give us something different to play for a few days before the expansion comes out, which will enable us to make a less difficult decision when it comes to picking which shiny new toy to play with. Hearthstone's Classic mode, and 20.0 patch, should launch prior to the pre-release weekend which takes place on March 26, though no time and date have been confirmed yet. We can expect to hear something during the final card reveal stream taking place on March 22.

Thanks to RidiculousHat for pointing this out.

Quote From Blizzard

This blog originally stated that Classic would go live alongside set rotation, which occurs at the launch of the expansion. We've updated it to reflect that Classic will unlock earlier, with the 20.0 patch.