Education Secretary Michael Gove apologised to MPs in Parliament for saying he had "welshed on the deal" (or he might have said "welched" - both spellings are acceptable). As the OED makes clear, "the word is sometimes considered offensive in view of the conjectured connection with Welsh people". These days 'welsh' is often followed by 'on a deal', but originally (1850s) it referred to dishonest bookmakers who didn't pay people their winnings. Welsh people were considered dishonest (some readers will have heard the old nursery rhyme"Taffy was a Welshman, Taffy was a thief' - it appears in an OED citation, too), hence the coining of the verb.
The word Welsh, referring to the people of Wales, has its origin in an Old English word for 'foreigner' and is cognate with a similar Old German word meaning 'speaker of a Romance language'. This is why French-speaking Belgians are called Walloons.
As for the old spelling Welch, the Royal Welsh regiment (with an s) was known until only 2006 (or part of it was) as the Royal Welch Fusiliers (with a c).
Here's a piece on the Michael Gove gaffe.