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There are two ways to do this.
POWER OF ATTORNEY
PROCESS: She signs a power of attorney giving you the right to make parental decisions on her behalf. See sample HERE.
PROS: No need to go to Court, immediate effect.
CONS: She can void the POA at any time. The father can override the POA at any time. While there is a presumption that the POA is valid, if she is ruled by the court or a medical entity not to have mental capacity at time of having signed it, the POA can be voided.
PROCESS: File in court to get formal custody - example of petition HERE. One would have to have her sign a waiver (assuming she has mental capacity) and serve the father, either directly or via substitute service if he cannot be found (newspaper, etc). Process can take 6+ months.
PROS: Custody cannot be voided by mother or father, they must file in court to modify back.
CONS: Court has to agree to this. Length of process. Process includes costly legal/attorney fees, evaluation of home placement, etc. Attorney not required but recommended.
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