It's a known letter, listed in his works with plenty of references to in on the internet. The " ink " almost looks like facsimile and there are thousands of books about Burns and other literary figures which sometimes include facsimiles of these letters " tipped in " to the binding.
" Burns also wrote to James Johnson in Edinburgh and requested a copy of "The Scots Musical Museum" for Jessie, saying "My wife has a very particular friend, a young lady who sings well, to whom she wishes to present the Scots Musical Museum, If you have a spare copy, will you be so obliging as to send it by the very first fly, as I am anxious to have it soon "
http://www.auldlangsyne.org.uk/letters/236.htmlThe paper stock also seems wrong for the period.
How did you acquire it ?