The client won't update yet though, you'll have to install it yourself.
It's been a year and a half since 'Minecraft' was first playable and the whole time that people have been investing massive amounts of their days into it, the game's only been in Alpha or Beta stages. Last night, the games creator, Notch, announced that 'Minecraft' is officially finished and that the second release candidate should be considered the final product. To quote from Notch's Twitter feed, "So, yes, Minecraft has gone gold :D"
As of this writing, the client isn't automatically downloading the patch, but you can go get it yourself and install it manually. Download the update from this direct link and then pop it into the proper folders. If you don't know how to handle the manual install, there's a handy guide at Minecraft Download.