Yes you can, but you will definitely need it later on. If you have a copy, please include with the submissions. USCIS does not usually ask for an original to be mailed.
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You can always file an incomplete application. However it will result in a delay in processing as you will get a request for evidence (RFE) seeking the birth certificate. Your adjustment application cannot be approved without the birth certificate or a legitimately documented non availability of it.
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I would agree with Mr. Segal that you probably can, but should take steps to request one from your country of birth because eventually establishing identity is your responsibility. In immigration law, identity is always a question.
The I-485 instructions say that if you do not have your birth certificate, you must demonstrate that either your birth certificate doesn't exist or that you can't obtain it, and that you must submit secondary evidence of your birth, such as church or school records that show proof of your birth.
Filing without it will cause, as the other attorneys mentioned, delays and problems.
Do things correctly ... not quickly.
Get the birth certificate before filing.
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