If she has had her green card long enough it may be wise to apply for citizenship. Otherwise you will have to wait, I would recommend speaking with an attorney, for a free consultation give me a call.
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Unfortunately marriage to a permanent resident does not give you the right to stay in the U.S. If your H-1B extension remains denied you may need to leave to preserve your options. You say she "just got green card" and since she wants to marry you we can conclude that her green card was not issued based on marriage to a U.S. citizen. Therefore it will be five years before she can apply for citizenship and probably another year before she is naturalized.
You need to considers what happens if your H-1B is approved and what happens if it is not approved. If it is not approved marriage to a green card holder may not be in your best interest. You may be best advised to wait to marry. A comprehensive immigration consultation will explore these and several other issues.
I think you are best advised to consult with an attorney before you make any final decisions.
I will agree with both of my colleagues. I just wanted to add something - Because you are out of status and because the greencard will not give you any status during the pending application. There is a chance that ICE can come and pick you up. It has happened for few of our clients. However, you might be able in front of judge to argue your case. I will recommend having a good lawyer working on the case to prevent any mishap.
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The only option for adjusting status through marriage is for spouses of US citizens. If she just got her greencard, then that option is not available to you. You should consult an attorney regarding your case and I would also keep informed about the impending changes in immigration law. Reform is on the horizon and there is always a possibility that part of that reform would benefit spouses of LPRs.