He undertook the difficult task. He did so albeit the fact that he did not have any spare time.

He undertook the difficult task in spite of the fact that he did not have any spare time.

He undertook the difficult task because of the fact that he did not have any spare time.

He undertook the difficult task contrary to the fact that he did not have any spare time.

He undertook the difficult task nevertheless for the fact that he did not have any spare time.