The Scotland international - signed for a reported £7 million from Norwich City during the close-season - was forced from the field seven minutes into the second half of Hull's Premier League opener against QPR on Saturday.
Hull won that match 1-0, but manager Steve Bruce expressed concern that Snodgrass could be braced for a lengthy spell in the treatment room.
And those fears were confirmed on Wednesday, with Bruce revealing Snodgrass would require surgery on the joint and describing the situation as a "tragedy".
"We've lost Robert Snodgrass and the diagnosis is he'll be gone for six months, which is a big blow to us," said Bruce.
"We spent a lot of money adding to our midfield area and now we've lost Snodgrass for the best part of the season. It's a tragedy for him and for us."
Snodgrass himself tweeted: "Gutted ....!!! But I will work as hard as I can to come back fitter and stronger."
Hull visit Lokeren for the first leg of their UEFA Europa League play-off on Thursday.
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