Death row cases are causing tension between Mexico and the US
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The US Supreme Court is hearing a case that could decide the fate of 51 Mexicans on death row who say that their rights were denied.
The International Court of Justice says the convictions break part of an international pact which allows foreigners help from their embassies.
Jose Medellin says he is entitled to a federal court hearing as a Texas court sentenced him without consular access.
But Texas says he did not raise his claim during his state trial.
As a result, it says, Medellin is barred under the US constitution from federal relief - and the state court judgement should stand regardless.
President George W Bush recently ordered states to comply with the ICJ ruling and hold new hearings for the convicted Mexicans.
"Whether the president has authority to issue such a broad determination is far from clear," Texas Attorney General Greg Abbott said in a recent filing.
US withdrawal
Medellin was one of five gang members sentenced to death in 1994 for raping and murdering two teenage girls.
His lawyers are invoking the Vienna Convention to get his sentence overturned.
Last year, the ICJ in The Hague ruled that the 51 convictions violated the 1963 convention, which was ratified by the US Senate.
Part of the Vienna protocol requires the ICJ to make the final decision when citizens of its signatory nations have been jailed abroad, specifically if they have been denied access to a diplomat from their own country.
Earlier this month, the US withdrew from a section of the treaty so that the ICJ could no longer have powers of enforcement.
The US state department said it was not appropriate that an international court should reverse the decisions of a country's criminal justice system.
Correspondents say the protocol was once seen as a means of protecting US citizens who had been jailed abroad, but in recent years the protocol has been used by opponents to the death penalty in the US.