"First secretary at the embassy of the Russian Federation in Abkhazia, Dmitry Vishernev, was murdered in a terrible criminal attack in Sukhumi," the Russian-backed territory's foreign ministry said in a statement.
"His spouse was badly wounded," the statement said, without giving further details.
Local law enforcement agencies have launched an investigation into the killing, which was aimed at "undermining the strategic and brotherly relation" between Moscow and Sukhumi, the statement said.
According to initial information, Vishernev was shot in the head in the garage of his home at around 8:30 am, state-run Russian news agency RIA Novosti reported, citing an unnamed representative of the Sukhumi police department.
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A spokesperson for the separatist authority's interior ministry told Russia's Interfax agency that evidence gathered at the scene indicated that the killing was a "terrorist attack".
Nestled on the Black Sea, the Russian-backed territory of Abkhazia declared independence from Georgia following a brutal war in the early 1990s.
Russia -- which for a brief war with Georgia in 2008 over South Ossetia -- has thousands of troops stationed in Abkhazia-- a situation condemned by the Georgian government as a de-facto occupation.
Georgia has been looking to improve relations with Russia and its own breakaway regions since the party of President Mikheil Saakashvili, a staunch US ally, lost parliamentary polls in October to a coalition headed by now-Prime Minister Bidzina Ivanishvili.