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Well, I believe there is also the problem of the correct type of needle/cartridge for the 78 grooves, right?
Correct. The groove in a 78 is wider and the stylus is larger. Most turntables with 78 RPM settings had a two-sided 'flip-over' stylus. Some higher-end (i.e. magnetic) cartridges had an optional 78 stylus that required you to remove-and-replace the stylus standard stylus when playing 78s.
I assume you've tried the yellow pages... If you can find a company that does vinyl-to-CD transfers, they should be able to do 78 as well as 45 & 33. Or, you might check with some places that do VHS-to-DVD transfer or film-to-DVD transfer... If they can't do it, maybe they know somebody who can.
And as a "last resort", you can still buy record players that play 78s. (You might be able to find a cheap one on eBay, and that might be the way to go if you're only going to use it once.) Most of the new ones I've seen for sale are rather low-end (cheap), but you'd probably get better results than using your high-quality 33/45 player (I'm assuming you have one).
Heartland sells an all-in-one tabeltop stereo with a 78RPM record player and a CD recorder. (Heartland sends me their mail-order catalog, but I've never ordered from them.)