Why not use the Rev-Ed AXE133? You can download the firmware straight onto a PICAXE-18M2.It is a handy little chip if you are trying to conserve I/O pins on your picaxe.
On this web page:http://www.phanderson.com/Does anyone know if this chip is still available anywhere? I don't see it for sale on the phanderson.com website.
In addition to the Rev Ed AXE133 mentioned by nick12ab, if you are only seeking a simple chip based solution there is also the rev Ed FRM010 serial LCD firmware chip:It is a handy little chip if you are trying to conserve I/O pins on your picaxe.