It's not public domain. Bill used to earn money from it when he was still alive. I would assume the rights have passed to his beneficiaries.

If you wish to record it, the process is pretty simple. You contact Mechanical Copyright Protection Society (MCPS)and apply for a licence. They will tell you how much you need to pay, and they will pass the money on to whoever is currently registered as the rights owner. Should it drop out of copyright at some point, then MCPS would issue a 'no payment required notice'.

As somebody else said, you do not need to get permission per se, just pay the money and you're good to go.

As an old friend of Bill's, I know that he would be chuffed to bits that his songs (his babies as he called them) still live on after him.