“I immediately burst into tears,” Catherine Han, 24, said of sitting in the audience and learning that her remaining three years would not cost her anything. “I feel like the pressure that students know was bearing down on me, and it just started to lift off. And I feel so light now because it’s really life-changing.”

Ms. Han, who is beginning her second year, said her father is a postal worker and her mother is a hospital technician. “I will be able to take care of my parents years faster than if this wasn’t offered, and that is such a burden off my mind,” she said. “I feel like I can focus on what I’m learning in the present and on becoming the best physician I can.”

Neil Kelly, also in his second year, said medical students who were less preoccupied by financial stresses can focus more on caring for patients. “It gives you that ability, moving forward, to not have to worry about the money playing a role in your specialty,” he said.

Mr. Kelly, 32, said his wife was expecting the couple’s second child. “It’s really difficult to make it work in the city on one income, and to have multiple children,” he said.

Financial concerns may deter doctors from practicing in communities where many patients are on Medicaid. Doctors have become scarce in many rural areas, where they often have trouble earning enough to pay off their debts. Some 113 rural hospitals have closed since 2010, according to the National Rural Health Association.

“While medical education remains an excellent investment for students, philanthropic support for medical schools’ financial aid programs can help recruit students from diverse and disadvantaged backgrounds who are more likely to practice in underserved communities,” said Matthew Shick, senior director of government relations for the Association of American Medical Colleges.

Weill Cornell had a scholarship endowment of $150 million before it announced a new $160 million gift on Monday. The school needs about $50 million more in the coming years to ensure that the program continues in perpetuity, Dr. Choi said, adding that he felt “very confident” it would reach that number.