US Birth Certificate
Any person born in United States is US citizen and proof of US citizenship is birth certificate for naturally born US citizens. Birth certificate is a very important document to get other documents such as US passport, visa etc.

When you go to hospital for child birth, you will be given the forms to be filled out by the hospital. There will be several forms. Among them, one form will be for hospital records and another to be sent to county or state.

Depending upon the state you live in, county/state form may either be directly sent to county/state or you may be asked to send that form to county/state. (e.g., in California, it will be sent by hospital to county, but in Virginia, you have to send the form to state yourself after 3-4 weeks of birth. )

It is best to think of baby's name well in advance as you will have to fill in the name before sending the form to county/state. It is also possible to add the name within short time after birth, but that is more work and not worth the hassle. If you are not sure whether the baby is boy or girl, you can think about name(s) of both boy and girl. If you believe that your baby's name should be according to some astrology or religous beliefs etc, you can call relevant persons before you leave hospital but get name written in the form before leaving the hospital.

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Hospital also will give record of birth but that is not too much valuable. It is temporary record and not actually valid for applyling social security number, passport, visa etc.

You can get as many original birth certificates you want in US. It is better to get extra original birth certificates and keep them with you. Depending upon the state you live in, there may be different amount of charge per copy for original birth certificate. Original birth certificate will be needed during different phases of person's life such as school admission, getting passport etc.

Birth certificates are normally issued by Vital Records Department of the State you are in.
State Department of Health Website