@vaimsarv - What is your exact MacBook model as there are a few different models which require different means.
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=== Update (05/24/24) ===
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This series doesn’t require any discharge procedure before hand, the MacBook series was unique in that regard.
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Just follow this guide [guide|89284]
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Once you disconnect the battery there is no risks to you but micro electronics is sensitive to ESD (high voltage static discharge) which it’s wise to use an antistatic mat which is grounded.
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As to the old battery, just dispose of the battery safely recycle it. While not ideal to have the battery charged it doesn’t pose a risk as long as you don’t create a short within it.