"despite the doubt" is correct and usable in written English.
You can use it to express that someone is doing something despite feelings of doubt or uncertainty. For example, "Despite the doubt he felt in his ability, he decided to take the opportunity and give it his best effort."
"We are assured that the business plan for the stadium remains robust despite the doubt involving Truro City FC and their commitment," he said.
The wage gap (which does actually exist and is especially obvious in Hollywood, despite the doubt that is created by the numerous and sometimes difficult to understand factors) could be closed and allow women to work the same jobs for the same amounts of money without trying to take anything away from the men.
Despite the doubt they might give my neighbors in regards to my mental health, visceral kinds of reactions like these remind me that, despite my struggles with the Catholic Church and our umpteen breakups over the years, I still care very much about it.
We'd definitely know about it if we had, despite the doubts of the conspiratorially minded.
Despite the doubts, President Bush's job approval rating stands firm at 87percentt.
Despite the doubts of others, Mr. Shatry said, "it will, it will".
There are, despite the doubts, now several operators offering longer routes at low prices.
Despite the doubts over the government's true intentions, the army is sticking to its official line.
Despite the doubts and grumpiness across the region, Mr. Chirac exhorted his countrymen not to fear necessary reforms.
Washing ton leaned on the International Monetary Fund to continue lending, despite the doubts of IMF officials.
Surely there was no way back, despite the doubts that must have been creeping into Federer's mind.
