Current was originally considered to be a flow of positive charge, and so it was labelled as flowing from positive to negative (the direction in which positive charge would flow). It was later discovered that it was actually the negative charges that moved, but for convention's sake they left it as positive to negative.
So yes, it is A to B. Current is always taken as flowing from positive to negative.
My question is: on the symbol for a battery, which is the positive end? The long thin line, or the short, fat line?