What type of stroke is caused by the rupture of a blood vessel on the brain's surface resulting in bleeding into the space between the brain and the skull?

A Hemorrhagic Stroke! 

It will leak blood into the subarachnoid space, an area between the brain and the skull.
They are usually caused by the rupture of an aneurysm, a bulge in a blood vessel, which creates a weakening in the artery wall.

Bis the correct answer.