7:56 AM PT -- King Charles will go into the hospital next week for treatment for an enlarged prostate, Buckingham Palace announced in a statement.

The 75-year-old's public engagements "will be postponed for a short period of recuperation."

Kate Middleton went under the knife to deal with something in her gut -- and now, the Royals are saying she's gonna be bedridden in the hospital for two whole weeks.

The Princess of Wales underwent a scheduled abdominal surgery Tuesday -- which Kensington Palace says was successful. They don't get into details about what the procedure was meant to address -- but they do say she'll remain hospitalized for the time being.

Specifically, they say Kate will be under a doctor's care for at least 10 days ... if not 14. They also say that as she recovers, she won't be able to perform her public duties as a Senior Royal until after Easter -- something they say was recommended as medical advice.

Again, Kensington Palace isn't getting into the specifics of what's going on with her -- and they explicitly ask the public to respect her privacy during this time. They note they'll only provide significant updates on her condition ... and nothing more.

The palace also says Kate apologizes in advance for anyone inconvenienced by her postponed engagements, and looks forward to getting back on her feet in due time.

While her ailment is a mystery, sources with direct knowledge have assured TMZ ... her condition is non-cancerous -- so we can rule that out right off the bat.

Still, it's a startling development, which seems to have come out of nowhere -- and more troubling is the fact that a 14-day hospital stay after a surgery is quite a long time. Mind you, we haven't seen Kate in public since December ... and she looked pretty okay then.

Here's hoping for a speedy recovery. Get well soon!