This type of malaria is a particular challenge to get rid of as it can remain dormant in the liver for years before reawakening many times.
Scientists have described tafenoquine as a "phenomenal achievement."
Regulators around the world will now look at the drug to see if they can recommend it for their populations.
Recurring malaria - caused by the parasite plasmodium vivax - is the most common type of malaria outside Sub-Saharan Africa.
Children can be particularly at risk, getting several bouts of malaria from a single bite, missing lots of school and getting weaker each time they get the disease.
And infected people can act as unwitting reservoirs of the disease because when the parasite reawakens in their bodies a mosquito can carry that parasite on to someone else.
This can make it hard to eliminate around the world.
Now the Food and Drug Administration (FDA) in the United States has given the seal of approval to tafenoquine, a drug that can flush the parasite out of its hiding place in the liver and stop people getting it again.
It can be taken alongside another medicine to treat the immediate infection.
There is already a medication that can be used to get rid of malaria hiding in the liver called primaquine.
But unlike the single dose of tafenoquine needed, primaquine often needs to be taken for 14 days.
Some experts are concerned that many people feel better after just a few days and stop taking the pills, allowing the parasite to awaken at a later date.
The FDA says the drug is effective and approves it for use in the United States but points out that there are important side effects to be aware of.
For example people with an enzyme problem, called G6PD deficiency, should not take the drug as it can cause severe anaemia,
The regulator recommends people are tested for the deficiency for this before it is given - which can pose a problem in poorer areas where malaria is common.
There are also concerns that at higher doses it can be a problem for people with psychiatric illnesses.
But despite these cautions there is hope the drug, together with bed nets and other precautions, will help reduce the amount of vivax malaria in the world.
Prof Ric Price, of Oxford University, told the BBC: "The ability to get rid of the parasite in the liver with a single dose of tafenoquine is a phenomenal achievement and in my mind it represents one of the most significant advances in malaria treatment in the last 60 years."
Meanwhile Dr Hal Barron, president of research and development at GSK, the company that manufactures the drug, said: "The approval of Krintafel [the brand name for tafenonquine], the first new treatment for Plasmodium vivax malaria in over 60 years, is a significant milestone for people living with this type of relapsing malaria.
"Together with our partner, Medicines for Malaria Venture, we believe Krintafel will be an important medicine for patients with malaria and contribute to the ongoing effort to eradicate this disease."
Tafenonquine has been in existence since the 1970s but working with Medicines for Malaria, GSK has repurposed the drug so that it can be used to get rid of malaria parasites in the liver.
The next step will be for the drug to be assessed by regulators in countries where this form of malaria is a significant problem.
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