I'll do my best to give you the 411 on demography:
Schools offering a PhD in demography are UPenn, Berkeley, and Princeton. They are all great programs, and their alum (from all 3 schools) have gone on to become some of the best demographers in the world. In fact, the Max-Plank Institute for Demographic Research (in Germany) and Vienna Institute of Demography are run by grads of Berkeley and Penn (respectively). At all of these schools, you can apply for a joint PhD in demography and another related discipline, usually sociology or economics. Also, if you are interested in public health/fertility, look at Johns Hopkis--PhD in reproductive health.
UTexas at San Antonio has started a PhD in applied demography, but I am not too familiar with its reputation and/or quality.
If social demography is your calling, you can receive great training in many universities' sociology phd programs. Wisconsin, Michigan, UNC, Duke, U-Texas, Penn State, UCLA, UMD, FSU, Brown, (I am sure I am missing some), and of course, Penn, Berkeley and Princeton.
Finally, the number of applicants for the PhD programs at Penn, Berkeley, and Princeton is FAR lower than the applicants for their sociology programs. It may seem like a great idea to apply to demography and then enter in sociology at a later time. If demography is your focus, consider this possibility, but also keep in mind that the applicants for demography are usually extremely qualified with both experience and/or master degrees. For sociology, there are a fair amount of undergrads applying directly for the PhD, likely making the applicant pool somewhat less competitive.
Any questions?
How I know this info: I am an aspiring demographer and will be starting a PhD program next year in demography