The 1991 murder of four teenage girls at a yogurt shop in Austin, Texas, confounded investigators and haunted the city for more than 30 years — until this week, when police said modern DNA investigative methods had helped them to at last identify a suspect.

The girls had been bound, gagged and set on fire inside the shop I Can’t Believe It’s Yogurt! on a commercial street in Austin, and the case came to be known as the Yogurt shop murders.

In a statement on Friday, the city of Austin said its police department had named Robert Eugene Brashers as a suspect. Mr. Brashers died by suicide in 1999, days after shooting himself during a standoff with police.

“Austin Police have made a significant breakthrough,” said the statement on the city’s website. “For almost 34 years they have worked tirelessly and remained committed to solving this case for the families” of the four girls, the city said. “We have identified a suspect in these murders through a wide range of DNA testing.”

The Austin Police Department said it would not offer more information on the case until Monday. The outcome of its investigation was first reported this weekend by The Austin American-Statesman.

Eliza Thomas, 17, and Jennifer Harbison, 17, had been working at the yogurt store on the night of Dec. 6, 1991. A friend, Amy Ayers, 13, and Jennifer’s 15-year-old sister, Sarah, had come to meet them there.