The tribunal was set up in 1988 under an agreement with the US when the Marshall Islands gained independence from its former colonial master.
Since 1991 it has paid personal injury claims of Marshall Islanders affected by 67 US atmospheric nuclear tests conducted in there between 1946 and 1958.
But the tribunal halted payments in 2006 due a lack of funds.
A Tribunal official said the compensation trust fund provided by the US has dropped from its original 1986 amount of 150 million dollars to just one million dollars, which is expected to run out by next year.
James Matayoshi, the mayor of Rongelap, said the islanders planned to begin a legal action against the US in the US Court of Claims to seek enforcement of the tribunal's 1.03 billion dollar award.
The award is the largest of the four "class action" awards made by the tribunal, none of which have been paid because of the lack of funds.
Bikini and Enewetak islanders began a legal action last year against the US government in the Court of Claims to get enforcement of their tribunal awards.
A hearing is expected to take place on April 23 in Washington.
The tribunal has already issued awards totaling over one billion dollars for claims filed by Enewetak and Bikini atolls, which were the sites of the 67 nuclear tests, and for Utrik, another atoll that was hit by fallout from the Bravo test.
Rongelap islanders were evacuated from their island home for three years after the Bravo test but were exposed to more radiation when repatriated to their island by the US in 1957, the tribunal said.
"Although the people were assured that it was safe to return to Rongelap in 1957, it was evident that the US knew Rongelap was still contaminated at that time," tribunal judges James Plasman and Gregory Danz, both Americans, said in their ruling.
The judges said that the people "came to feel like guinea pigs, used for experimentation by the US."
Rongelap Islanders left from the atoll in 1985, fearing continued radiation exposure.
The islanders "suffered emotional distress and a degraded quality of life as a consequence of the contamination of their property," the ruling said.
The award covers loss of property value from radiation contamination, the costs of a clean-up to allow future resettlement, and hardship and suffering.