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Unlike other Australian visas, the ETA is not transferable to a new passport. If you get a new passport, you must apply for a new ETA with the new passport.
You should also be aware that the six month passport validity rule does not apply to Australia; your passport only needs to be valid for the duration of your journey. This means you do not need a new passport before your travel to Australia next month. You can renew it after you return home.