Wordorigins.org

Welch is an archaic form of Welsh, one that pretty much only survives in the sense of to fail to pay out on a lost wager. This usage dates to around 1857 when it appears in the 8 June issue of the Morning Chronicle:

He got his living by “welching” and taking in the “flats.”

The origin is obscure, but it undoubtedly originated as an ethnic slur of the Welsh people.

(Source: Oxford English Dictionary, 2nd Edition)