` My 10 year old iPOD will not hold a charge for more than 45 minutes.

While the battery can be replaced it will not be able to be done by Apple.

You will have to find a third party technician if you wish to have it done. The cost of finding a newly manufactured battery as well as a technician who is able to replace it may be more expensive than it's worth.

Apple will replace the battery or give you an identical model refurbished iPod with a new battery for $79US plus shipping. See: https://support.apple.com/ipod/repair/service

I did see that. The OP probably has an iPod 4, which, as far as I can tell from the link I provided is "an eligible model" for a battery replacement.

That would typically be the case except you failed to notice that the user states that their iPod is 10 years old.