彼が病気だという

that he is sick

→ 彼
phrase Rare #37,715
that he is sick Introduces a statement or belief, often before こと, の, 事実, or another noun. The という part marks the preceding clause as quoted or described content.
JA: 彼が病気だという話を聞いた。
EN: I heard that he is sick.

彼が病気だというのは一体本当だろうか

I wonder if it's really true that he's sick.

彼が病気だという事実は彼女にはそれほど大したことではなかった

The fact that he was sick wasn’t that big a deal to her.

彼が病気だというのは明らかなようだ

It seems obvious that he is sick.

つい彼が病気だということをしゃべってしまった

I accidentally let it slip that he was sick.