Battery charging problem.

[Isaiah92260]

So the battery I have in my 63 is a year old according the the previous owner. The Alternator looks pretty old, I don't know know how old but I can tell it has some age... Anyways, my battery keeps doing at least once a week after I give a charge over night. The charging light on my dash stays on the whole time I'm driving, it's on pretty dim, but bright enough to see during the day. So I'm thinking that's saying that the alternator isn't gettinf 100% charge. So does anyone think Its just time for a new alternator or I'm or have a poor ground or connection somewhere? I can provide poi tires of whatever anyone may ask to help.