Washington ? Try fewer burgers and more veggies after menopause: Cutting dietary fat may offer a long-sought protection against deadly ovarian cancer – if you stick with the diet long enough.

Low-fat diets have long been promoted as a way to reduce the risk of different cancers, with decidedly mixed results when put to the test.

But Tuesday, researchers unveiled the first hard evidence that switching to a low-fat diet late in life can lower the odds of ovarian cancer, a malignancy with a particularly dismal survival rate.

The study tracked almost 40,000 women ages 50 to 79, some of whom were assigned to cut the total fat in their diets to 20 percent of calories – from an average of 35 percent – while others continued their usual diets.

For the first four years, the menu changes didn’t make a difference. But those who kept the fat low for eight years cut their chances of ovarian cancer by 40 percent, researchers reported in the Journal of the National Cancer Institute.

“This is really good news,” said Dr. Jacques Rossouw of the National Institutes of Health, which funded the work. “But you have to stick with the diet.”

Until now, the only known prescription against ovarian cancer – aside from surgically removing the ovaries – was for women of childbearing age to use birth control pills. Use for five years can lower the ovarian cancer risk by up to 60 percent, protection that lingers years after pill use ends.

The new findings offer an option for postmenopausal women to try.

It’s arguably the most promising finding of the mammoth Women’s Health Initiative dietary study, which enrolled tens of thousands of healthy women to track the effects of teaching them to cut fat and eat more fruits and vegetables.

So far, the diet has had seemingly little impact on rates of breast cancer, colorectal cancer and even, surprisingly, heart disease. There are a number of theories: Maybe the women started healthier eating too late; most were overweight, a major risk factor, and the diet wasn’t designed to shed pounds. Nor did most women actually cut enough fat.

Despite all those hurdles, a low-fat diet did appear protective against ovarian cancer – and the women who started with the worst diets and cut fat the most got the most benefit.

Ovarian cancer is fairly rare, affecting one in 60 women compared with the one in 9 who will get breast cancer. But it is particularly lethal because it usually is detected only after it has spread throughout the abdomen, making it much harder to treat. Only 45 percent of patients survive five years.

The American Cancer Society estimates that 22,430 U.S. women will be diagnosed with ovarian cancer this year; 15,280 women will die of it.

Ovarian cancer can strike anytime in adulthood, but risk increases with age. Mutations in the so-called breast cancer genes BRCA1 and BRCA2 also increase the risk of ovarian cancer – and women in the new study have not yet been tested for those genes, to see whether the low-fat diet proves more or less beneficial for them.

Why would diet affect ovaries? The theory is that fat intake increases the amount of estrogen circulating in the blood, which may in turn overstimulate sensitive ovaries.