The 51-year-old royal family member underwent emergency surgery on Friday at Toulouse University Hospital in southern France.
Joachim, 51 and sixth in line to the Danish throne, was holidaying with his family at Cayx, a chateau the royal family owns in southwestern France, when he was taken ill.
“After the recent days of hospitalisation and treatment in the intensive care unit, it is the doctors' assessment that Prince Joachim will not have any physical effects or other injuries as a result of the blood clot,” the palace said in a statement.
The prince was expected to be moved out of the intensive care unit shortly, but would remain in hospital “for some time to come”.
“The blood clot in the brain was due to a sudden dissection of an artery, and the hospital's medical team assesses that the risk of recurrence is very small when the artery has healed,” the palace said.