A child born here is a citizen automatically, by birth. The parents, in turn, can request residency on the grounds of being parents of an Argentine citizen.
The child needs to receive a birth certificate, then a DNI. Follow
this link to get an appointment at a Registro Civil office.
If you are not fluent in Spanish, you will have to be accompanied by an official translator.
IIRC the process will be expedited if you request only the birth certificate when registering the birth,
not the DNI.
The newborn DNI is free; for a 600 peso fee, this can be done within 24 hours.
Passport can be done once you have the DNI, within 48 hours. Passport costs 950 plus 1150 expedited fee.
Child can be on his/her way within 2 weeks of birth, which is also the minimum age at which most airlines will transport a newborn.
Once child has birth certificate/DNI, parents can request an appointment for obtaining permanent residency. Currently, turnos are being given 10 days forward.
The list of requirements is listed
here and includes: passport; DNI and birth certificate ("legalized"; no idea what that means) of the child; criminal background check from home country; criminal background check from Argentina; proof of address. Note that any documents from abroad must be legalized (either by Apostille or in Argentine consulate with jurisdiction where the documents were issued) and translated to Spanish by an official translator, which translation must also be legalized.
I know of a couple of people who help out with this for a fee, PM for details.