Giving up parental rights

The query is short of material facts.

How is that the mother is "can not care and is not interested" or "unable to or unfit to or unwilling to care for the child"?

Is it that "her parental rights and give complete custody to the other parent" ofr just an assumption?

What is the present status of the relationship among the parents  - are they divorced or judicially seperated?

If father is willing to take full custody of the child and the mother has no objections to it in a matrimonial dispute, then the court can pass an order favoring the husband 

There is no ambiguity in care and custody of minor by his/her father. 

Thank you experts, the reason is an out of wedlock child in this case and the mother wants to avoid social stigm and move on to a different marriage as per her parents wishes etc. Father wants to take full responsibility and raise the child, baby is 1 month old.

If its legally possible to give up parental rights so Father can take full custody of the young baby, then it solves the issue.

From above replies, it does appear this is possible to do through a qualified advocate. 

Your understanding is right.

The father can undertake the responsibility and the custody of the child if the mother is afraid of the social stigma, but remember that both the biological parents cannot abandon the child, it is a punishable offence

Dear Sirs, Any one in Chennai can help with drafting of taking up this case? I believe it will be in Family Court.

thanks,