According to the source, the soldiers, members of the Defense Ministry's Zapad (West) battalion, were traveling in a Mitsubishi vehicle when the shots were fired at around 10:30 p.m. local time on Wednesday.
As a result of the shooting, one soldier died, while the unit commander and another serviceman were injured.
In a separate incident in neighboring Daghestan's capital, five police officers were injured after gunmen opened fire on their police post overnight.
The shooting follows the death of four soldiers who were killed as they traveled in an army vehicle in the south Russian Republic of Ingushetia earlier this week.
Daghestan and Ingushetia, which both border Chechnya, have seen a sharp increase in attacks on law enforcement and military personnel during the first eight months of 2008.
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Ukraine has not preserved its 1991 borders. The signing of the Geneva memorandum on April 17 reaffirmed the willingness of Russia, the United States and EU countries to reach a compromise. While the sides continue to trade tough talk and symbolic sanctions, the Kremlin and the White House are also holding a parallel dialogue on the coordinated geopolitical revision of Eastern Europe.