LYNCHBURG, Va. — Here in the postpartum ward at the Virginia Baptist Hospital, vital records secretary Joann Butler has begun asking unwed fathers a question that used to be put to them later on, in a courthouse, by a judge: Do you want to acknowledge paternity?

A surprising number of unwed dads take a look at the little bundle of joy they have helped bring into the world and sign on the dotted line.

"We've finally figured out that the golden moment is right after birth," said Larry Jackson, Virginia's commissioner of the Department of Social Services, which is behind the program to move the process that establishes paternity out of the courts--where it's often been a time-consuming, costly, low-yield chase--and into the hospitals, where it can be settled in a matter of minutes. "That's when the dads are bursting with pride," he said.

"If you wait six months or a year later, the relationship (between the mother and the father) may very well have deteriorated and there are going to be problems," said Robert Krause, district manager of the state's child support office in Lynchburg.

"Problems" is a polite way of saying that unwed fathers as a group are notoriously difficult to track down and serve with child-support orders. Just 23.9% of never-married mothers nationwide received child support from absent fathers in 1989, compared to 72% of divorced or separated mothers.

Moreover, never-married mothers head the fastest-growing and the poorest families in the country; 26% of all births in the United States are out of wedlock, four times the rate of a generation ago. Last year, 54% of all never-married mothers were poor, partly because they received little or no child support.

If there is a silver lining to these demographic trends it is that as unmarried parents have become more numerous, they have become less stigmatized, so fathers are more inclined to acknowledge paternity. In 1989, when there were more than 1 million out-of-wedlock births nationwide, 335,589 paternities were established--up from 231,838 of about 831,000 such births in 1985.

"Lots of unmarried fathers do want to do the right thing and acknowledge paternity, but until last year, we forced them to go into court, sometimes two or three times," said Harry Wiggins, director of Virginia's division of child support. "For a lot of these young, low-income fathers, courts don't necessarily have the best association."