Rep. John Barrow has avoided another primary threat. A potential Democratic Party rival, state Sen. Lester Jackson, said no to the party leaders who were urging him to run because they’re upset about Barrow’s vote against the new health care reform law.

Jackson, who had been invited to the White House when the health law was signed, said Monday that he will run for re-election instead of challenging Barrow.

Barrow faces former state Sen. Regina Thomas, who is black, in a rematch of the 2008 primary, which Barrow won with ease after then-Sen. Barack Obama’s endorsement.

Some Congressional Black Caucus members have indicated that they’re not inclined to give Barrow as much help as they did in 2008, when they helped him win in a district that is 44 percent black.

Thomas raised $10,000 in the first quarter and had just $4,000 in the bank on March 31. Barrow raised $205,000 in the first quarter and had $825,000 banked at the end of last month.

“Has [Barrow] dodged a bullet yet? Not until the primary is over,” Chatham County Democratic Chairman Tony Center said. “It depends on whether the people who are unhappy with the health care vote stay motivated.”

The district’s county chairmen organized a conference call earlier this month to discuss their unhappiness with Barrow, but nothing has come of that discussion, Center said.