That year, at the Democratic National Convention, the state's delegation cast 15 votes that pushed Sen. John F. Kennedy over the top and made him the party's nominee for president.
With this year's race for the Democratic nomination still unsettled after primaries this week, Wyoming Democrats are feeling relevant again as Sens.and focus on the 12 delegates at stake in Saturday's caucus.
Only in the last few weeks have the campaigns stepped up their presence in the state, opening offices and calling voters and sending mailers. The first visits come Thursday, when former President Clinton makes three appearances. Obama and Hillary Clinton follow, each scheduling two events on Friday.
Clinton plans town-hall meetings in Casper and Cheyenne. Obama, who has had greater success with caucuses, plans a rally in Laramie at the University of Wyoming.
On Saturday, the state has 23 county caucuses
All the attention has Wyoming Democrats feeling giddy.
"It's amazing," said party spokesman Bill Luckett. "It's a once-in-a-generation type of opportunity for Wyoming to have a say in this presidential race."
The number of registered Democrats has grown by more than 1,000 since the start of the year, the secretary of state's office says. Republicans have 136,000 registered voters, compared to 59,000 Democrats.
"In the grand scheme of things, Wyoming factors in for us," said Obama spokesman Matt Chandler. "We take it very seriously."
Clinton spokesman Blake Zeff said potential volunteers are reaching out. "We're going to be harnessing that enthusiasm to really bring out the vote," he said.
It remains unclear which Democrat might have an edge; no public polling has been done in the state. However, at least one prominent Wyoming Democrat has questioned whether Clinton's unpopularity would hurt other Democratic candidates in the West.
"I feel a lot of conservative Republicans may not show on Election Day to vote for Sen. (John) McCain, but they may show up to vote against her," John Millin, the state Democratic Party chair, said of, the Republican nominee-in-waiting.
His comments sparked some controversy in the state.
"I think Chairman Millin is wrong. And I would go further. I would say it ill behooves the chairman of the Wyoming Democratic Party to rule out one of the two leading contenders for the office," said Kathy Karpan, who was in charge of an Interior Department office during the Clinton presidency.
Karpan, who also was Wyoming's secretary of state, said Hillary Clinton has had "every aspect of her political life scrutinized" while less is known about Obama.
Meanwhile, the state's top Democrat, Gov. Dave Freudenthal, has said he doesn't like any of the candidates, Republican or Democratic, because they haven't spoken enough about Western issues. He hasn't endorsed anyone and will not say who, as a superdelegate, he will vote for at the national convention.
"I just don't usually get involved in it. I may or may not," Freudenthal said.