It's the battery, if it were discharging through the alt due to defective diodes it would not be anywhere near as quick to do so as this.
I would clean up all connections anyway, as the batteries age the plates break up and the broken bits fall to the bottom causing a permanent short in that cell, years ago some were washed out then refilled and got a bit more life out of them, but most are sealed these days.
I always use Varta Blue myself, expensive but get what you pay for.
carbatteryshop.co.uk is a good site
tabetha