'but despite the fact that' is correct and usable in written English as a subordinating conjunction. You can use it when you want to introduce a statement which contrasts with the idea just mentioned. For example: Even though we had planned the event for months, but despite the fact that the weather was bad, the turnout was still huge.
"But despite the fact that he has no respect, the government has respect for him".
But despite the fact that these are both ways of knowing, some consider the difference between them to be significant.
It's freezing, but despite the fact that she's wearing no hat, she doesn't seem to notice the chill.
But despite the fact that Hollywood moved on years ago, this city of spas is hardly dead.
But despite the fact that a two-year CD was paying a lot more than the zero she was earning, she didn't want something new.
But, despite the fact that Detroit is now more than eighty per cent black, the area has never produced a major African-American hip-hop star.
"It was a very difficult time for me personally, but despite the fact that we were living apart, Alex was always there for me," Ms. Dineen said.
Ludwig does not simply clarify my doubts with English writing, it enlightens my writing with new possibilities
Simone Ivan Conte
Software Engineer at Adobe, UK