Michael Ikenna Nduanya, 34, and his common-law spouse Nguyen Thi Hai Anh, 27, were convicted on Tuesday in Ho Chi Minh City, the official said, refusing to be named.
He said the couple were arrested in December 2009, but gave no further details.
The website of Tuoi Tre newspaper, citing the indictment, said Nduanya had asked his wife to collect about one kilogram (2.2 pounds) of heroin from Cambodia. She was arrested at a Ho Chi Minh City bus depot after returning with the drugs, it said.
The report suggested the heroin originated in India and was ultimately destined for China.
Nduanya is the third person to be sentenced to death in Vietnam this year, according to state-controlled media reports tallied by AFP. There were 80 death sentences and four executions last year, according to the tally.
Most death sentences in the communist country are issued for drug and murder cases.
Under a decision by Vietnam's National Assembly, execution by lethal injection is to replace the firing squad on July 1 this year
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