not sure such a bank exists, but the canadian exchange (virtex.ca) seems to be close in that your account is similar to a bank account in that you can transfer both BTC and CAD into and out of your account - directly into your BTC wallet or your CAD bank account easily, *relatively* quickly (1 or 2 banking days), and *relatively* cheaply (3 bucks per withdrawl for CAD, 0 on BTC)

they *seem* to have learned from the mistakes of other exchanges, and perhaps their model should be mimicked in other countries, but only time will truly tell...

*edit* but the premise of the thread regarding making bitcoin more "user friendly" holds true. having to un-encrypt my wallet, open the client, make the exchange, and then re-encrypt my wallet and "delete" the original file *(who knows if it's ever truly deleted, even after emptying the recycle bin? not to mention that the whole time I'm trying to catch up to the network my wallet is pretty much fair game)*

if this doesn't get easy enough for the masses, it will never be adopted by the masses.