I recently got a free acoustic guitar. Although the body is about the same size as my other acoustic, the neck is quite shorter (pretty sure it's a kid's acoustic :P). Is it possible to replace the neck of an acoustic guitar? How much would it cost? How would I go about doing it?

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It would probably be cheaper just to buy a new guitar. The process of replacing an acoustic neck, even if it is a bolt on, is presumably much harder than doing so on an electric. The guitar is likely lower quality in the first place, and I know of a number of good full size import acoustics for <$300 that would be better and likely quite a bit cheaper than custom making a scale conversion neck that would work on that guitar.