The volcano, Mt. Kusatsu-Shiranesan, is located near a popular ski resort, and the people injured were skiing on the slopes and hit by flying rocks.
Six of those trapped were members of Japan's Ground Self-defence Force engaged in winter training manoeuvres, Defence Minister Itsunori Onodera said.
The agency said the eruption and the avalanche could not be linked immediately.
The falling rocks kicked clouds of snow into the air and they made impact.
Footage broadcast on Japanese television showed thick black smoke interspersed with falling rocks sliding down the snow-covered side of the volcano towards a ski slope. "Volcanic rocks rained down on the roof of the gondola". Images showed the windows of the gondola had shattered.
Eight of the people injured were soldiers conducting drills for rescue operations nearby, Associated Press reported.
A number of people were injured following the eruption of Japan's Mount Kusatsu-Shirane volcano early Tuesday.
Kenji Nogami, a professor of geochemistry at Tokyo Institute of Technology who has been surveying the mountain, said the latest eruption was a "typical phreatic eruption", in which groundwater heated by lava turns into vapor and explodes. Local police said two others had broken bones and were in critical condition.
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Bevin said earlier in a statement, "This is a tremendous tragedy and speaks to the heartbreak present in our communities". They always had a smile on their face and gauge, tge, the one that got injured pretty bad, he is a good buddy of might.
In the afternoon, Chief Cabinet Secretary Yoshihide Suga said one person died and 11 others were injured, and that the government has not learned of additional casualties or people missing.
Officials said they have not yet determined if the volcanic eruption caused the avalanche, but they did happen simultaneously, according to the Sydney Morning Herald.
A soldier was killed and at least 14 more were injured yesterday when a Japanese ski resort was targeted by Kusatsu-Shirane.
Japan has 110 active volcanoes and actively monitors 47 of them.
The site of Tuesday's blast last erupted about 3,000 years ago, scientists say.
Tuesday also witnessed another volcanic eruption in the Philippines, an natural disaster in Jakarta, Indonesia, and another quake off the coast of Alaska.
In September 2014, 63 people were killed on Mt. Ontake, on the border between Nagano and Gifu prefectures, in the worst volcanic disaster in Japan in almost 90 years.