The shooting, which left more than a dozen other soldiers including a senior German officer wounded, rocked the US-led project to train up the Afghan army as NATO combat forces withdraw after 13 years of fighting the Taliban.
The Afghan soldier was himself killed after he opened fire during a high-level visit by NATO officers to the Marshal Fahim National Defense University, a sprawling training complex on the outskirts of the capital.
Kirby said that the fatality was the highest-ranking US officer to have been killed since the 9/11 attacks when Lieutenant General Timothy Joseph Maude was killed by a hijacked airliner that crashed into the Pentagon.
No US general has been killed in combat since the Vietnam War.
Kirby said that "up to 15" people were injured in today's attack. Separately, the German army said that one of its generals was wounded.
"We believe that the assailant was an Afghan soldier," Kirby added.
The shooting was by far the highest profile "insider attack" of the Afghan conflict, in which scores of US-led NATO troops have been killed by Afghan soldiers turning their guns on their allies.
Also known as "green on blue" attacks, the killings have bred fierce mistrust among soldiers and forced joint patrols to be overseen by armed guards.
The Taliban did not immediately claim responsibility for the attack, and Western officials say that most such attacks stem from personal grudges and cultural misunderstandings rather than insurgent plots.
President Hamid Karzai condemned today's attack as a "cowardly" strike against Afghan and NATO officers.
"It is the work of those enemies who do not want to see Afghanistan have its own strong institutions," he said.
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