Actually hydrogen makes up the majority of the mass of a star throughout its entire life, even during helium fusion. Basically, things proceed like so:
Core hydrogen fusion produces the energy that powers the star for most of its lifetime.
At a certain point, all of the hydrogen in the core has been burned. Note that hydrogen is fusing in a shell around the helium core.
The helium core, since it isn't fusing -> not producing any outward force collapses inwards. As it collapses, it grows hotter and hotter, but there still isn't any helium fusion. The heat produced by this collapsing core actually causes the shell hydrogen fusion to increase like crazy. This extra hydrogen fusion causes the star to become a red giant.
At a certain point, the helium will become so hot and dense from its continued collapse, that it will finally begin to burn (~10^8 K as Andrew said.) This causes the helium core to expand, and the hydrogen fusion becomes confined to the core of the helium region. This actually cools the hydrogen shell by quite a bit, causing hydrogen shell fusion to decrease tremendously. This causes the entire core-shell output to drop, and the entire star becomes somewhat smaller as it collapses inwards. Now the star is running almost entirely on helium fusion.
This process is known as "helium flash."
But at the end, you are left with a start that is nearly entirely made of cooler hydrogen, with a small and extremely hot ball of helium at its core.