Step 1: Nitrobacter is a chemoautotroph that gains energy only by oxidising nitrite to nitrate.
Step 2: This oxidation step uses \( \text{O}_2 \) as the terminal electron acceptor in its electron transport chain.
Step 3: Without oxygen, the enzyme cannot function, so nitrate cannot form from \( \text{NH}_4^+ \).
Step 4: Both statements hold, so the correct answer is \[ \boxed{\text{Both Statement (I) and Statement (II) are correct}} \]