The simple answer is that your husband should apply for a "Consular Report of Birth Abroad" soon following your child's birth. Your husband should make an appointment with the US consulate that services the city your family lives in abroad. Since the child is born after 1986 there is a modified physical presence required for the US citizen parent--in this situation your husband must have continuously resided in the United States for 5 years and 2 of those years must have been after the age of 14.
Following this message there is a link to the documents needed for a CRBA from the US embassy in Mexico (that is the most common country most of our clients inquire regarding birth abroad to a US citizen parent but check the US embassy/consulate website that serves the specific foreign country you reside in).
Because each family's situation is different the information here is general in nature and you can always contact the US Embassy/consulate near you regarding specific issues.
Good luck and best wishes.
The information is not legal advice and informational only. No attorney client relationship should be construed.
It depends in how long your husband has been a us citizen. If he was born a citizen or has been one many years, then do the report of birth abroad of citizen. The other concern is you maintaining your residence, since simply flying back once a year is not sufficient.
This reply is intended only as general information and does not constitute legal advice in any particular case. This reply does not create an attorney/client relationship.
The best thing to do, if it can be arranged, is for you to come back to the US and give birth here. Then, there would be no issue as to whether the child is a native born US citizen, especially if, later on in the child's life, the child runs for high public office.
If you fly once a month to the US, you may not be maintaining your status as a legal permanent resident. If you do not seek permission from the US government to spend more time outside the US than inside, the government may one day decide to strip your status as a legal permanent resident.
If you give birth abroad, the US Embassy/Consulate of the country where you give birth has information on what US citizens must do to get documentation that the child is a US citizen.