majenko:
1: yes.2: Whatever the battery requires at that moment in time.
With a power source only one of either the voltage or current is "constant" - i.e., fixed at a specific value by the supply - the other value is free to wander around to whatever the connected device requires it to be. Normally it's the voltage that is fixed, and the current is determined by the circuit attached to it. With a constant current source the current is always the same, but the voltage is set by the circuit attached to it.
That said, there will be an upper limit that the supply can give for the "variable" value, and this has to be taken into consideration (i.e., you can't draw 40A from a 1A, 12V power supply or it will melt). So for a constant current supply you have to know what the upper voltage is likely to be and cater for it. With a 12V battery that's likely to be as high as 15V while charging, plus you have to factor in the 1.25V drop of the LM317T. So you are likely to need at least 17V going in to your current regulator - better to be safe and have a few volts more. A good choice would be an old laptop power supply as these are typically 19V.
However, there is one more gotcha in all this. The LM317T is liable to get VERY hot. You will need a Big Ass™ heat sink on it. Personally I would go for the LM338T as it has a mich higher current handling capacity and shouldn't get quite as hot. You'd still probably need a heat sink though, but not quite as big.
Thank you for your reply, i just have a few questions.
Firstly i do not understand how the voltage will change when charging. Say for example i have a flat battery and i put it on a constant current charger, will the voltage be highest or lowest and why? also if the battery is fully charged and i connect it the charger will the voltage be at highest or lowest? why does this voltage vary? does the battery act like a variable resister and voltage across it changes?
Secondly the heat that my LM317 will need to dissipate into the atmosphere is calculated how? for example if i use a 20v power supply and say for argument sakes the battery is charging at 13v. then the heat would be (20-13)*0.625A = 4.2W? and if i understand correctly the voltage at which the battery will charge will vary through the charge cycle is that correct?