The Bureau of Vital Statistics in Santa Fe, New Mexico issues birth and death certificates for the state. They can be contacted at:

Office of NM Vital Records and Health Statistics
1105 S. St. Francis Drive
P.O. Box 26110
Santa Fe, NM  87502-6110
(505) 827-0121

More information can be found at their web site. 

Birth certificates are also issued in Las Cruces:

Dona Ana County Health Center
1170 N. Solano, Ste. A
Las Cruces, NM 88001

Apply Monday-Thursday:  8:00 AM-11:00 AM, 1:00 PM-4:00 PM
Friday hours: 8:30 AM-11:00 AM; 1:00 PM-4:00 PM 

Charge is $10.00 with valid government issued photo identification.

You can also check here for addresses for requesting out-of-state birth certificates.