If you mean the disease itself, yersinia pestis it didn't. The Plague still breaks out now and then. For example, it killed 52 people during an epidemic in Surat, India in 1994.
As some of the other commentators have said, Medieval Europe really didn't end the Plague/Black Death. Even after the huge epidemic around the middle of the 14th century petered out, it still regularly popped up in Medieval Europe, ever decade or so, with declining rates of infection and lethality.
With each successive decade, local rulers did get slightly better at handling a re-occurrence. Even if they did not understand how the disease was spread, subsequent outbreaks were met with enforced quarantine and could be very Draconian. There are recorded instances in Italy of plague sufferers and their entire families being walled into their houses and left to die. Other cities would simply ban visitors, or ships calling at port in hopes of keeping carriers away. Practices such as these, along with the fact that the Pneumonic strain of the plague came around a lot less frequently meant that additional outbreaks tended to kill less people.
Source: A Distant Mirror: The Calamitous 14th Century by Barbara Tuckman
There are many good answers already and I would like to mention the great influenza outbreak of 1918 to further the idea that we are still at risk as a species for plague related die offs.
Consider that modern understanding of infectious diseases changed how the word plague is defined. Now it is rather narrowly considered by the general public (at least here in the United States!) to mean the Black Death. But in the past, the plague ascribed to be all manner of highly infectious diseases.
In the late 1910s and onward, influenza infected 20-40% of the entire population of Earth. The low end estimate was that fifty MILLION died.
Plagues aren't eradicated in any sense.
An estimated one third of the world's population (or ≈500 million persons) were infected and had clinically apparent illnesses (1,2) during the 1918–1919 influenza pandemic. The disease was exceptionally severe. Case-fatality rates were >2.5%, compared to <0.1% in other influenza pandemics (3,4). Total deaths were estimated at ≈50 million (5–7) and were arguably as high as 100 million (7).
Approximately 20% to 40% of the worldwide population became ill. An estimated 50 million people died.