What steps should I take to transfer my visa stamp to a new passport while in Thailand?

I just did mine when I renewed my marriage visa didn't cost me anymore I just paid for my marriage visa

Your US Embassy will provide a letter to the RTP Immigration to transfer your Visa Stamps.

Take this letter, your old passport and new passport to IO and they will transfer the "Visa stamps" for you, for whatever fee is charged.

But as Sebastian has said bring all your documents with you, have them photocopied, sign each copy with blue ink pen. (Even take a copy of your new passport application, you won't need it but you could be asked for it including the receipt, maybe even the envelope it came in...) The IO's basically can ask you for anything!

US side will give you a letter for you to submit to Immigration with your 2 passports, old and new.

I have seen immigration asking the receipt of your new passport… crazy but bring all documents just in case.

I went to IMMI and they took care. 10 min and free. You need a letter from Embassy, sometimes not, sometimes they ask.

Best thing to do is contact your embassy. But normally they transfer into your new passport at immigration.

Take your old and new passport to immigration, they will transfer your stamp to new passport for 500 baht