Hello. My name is Marc. I'm an experienced attorney and I will be happy to answer your questions.
I can understand your anxiety as to whether your conviction might impact your immigration status. The answer is - it might cause you problems, but I think the odds are in your favor.
Unfortunately, immigration authorities deny green cards to those with drug convictions. In fact, even for those who already have green cards, if/when the feds find out about an old conviction, the person is subject to deportation. I know because I used to be a prosecutor who dealt with green card holders in this situation.
One thing in your favor is that marijuana is not taken as seriously as narcotics. Another thing in your favor is that you were not convicted of sale or possession with intent to sell. If the amount of marijuana was more than one pound, however, it might be a problem. But - the only way the feds would know the quantity is if it was mentioned during your plea in front of the judge.
So, what does all this mean?
First - if your paperwork is already filed with immigration, they will most likely not find out about your conviction before making a decision.
Second - if they do find out and IF there is no evidence that you possessed a large quantity of marijuana, they will likely approve your application anyway.
Third - if your application is denied for this reason, you have the following option, for which you'll need a lawyer:
You can ask the court to vacate your conviction. Even though you entered a plea of guilty, the court will toss the conviction out if you can show that you received "ineffective assistance of counsel" - meaning that your criminal defense lawyer screwed up.
How can you establish that you received ineffective assistance of counsel? By telling the court that your lawyer failed to explain to you that your guilty plea might have negative immigration consequences. This issue has been well settled in the courts - if an alien is not told about the possibility of immigration consequences BEFORE he pleads guilty, the alien's conviction can be vacated.
Should the court grant your request and vacate your conviction, you could then apply again for a green card with a clean record.
Be advised, though, that when a conviction is vacated, the prosecutor has the right to make you go to trial. In my experience, however, prosecutors rarely do this - especially for nonviolent crimes and especially not for marijuana charges.
So overall, you should be ok. Hopefully you won't have to use the third option, which is time consuming, but effective.
I hope my answer helps you to better understand your issues and options. If so, please be sure to rate my answer, since that is the only way I can receive credit.
I wish you luck and hope you'll receive your green card soon!