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I would work directly with the judge at the state court and I would work under the supervision of a law clerk at the immigration court but the immigration court is small and there are two judges who I will also be working with. Immigration court is part of DOJ. Which one is more prestigious?

neither?

What do you want to do and what kind of experience do you want?

I assume when you say "state court judge," you're talking about a trial court. So its either a criminal or civil docket or an immigration only docket. EOIR internships can really help if you want to do EOIR upon graduation. If you don't want to do immigration, you're not going to learn very much. Most immigration matters will be somewhat routine, and you can't make constitutional rulings. You pretty much just apply facts to law based on the plain langauge of the INA and BIA precedent. If that doesn't sound like something that will be helpful and/or interesting, I'd recommend working with the trial judge. At least there you'll get to rule on motions, read lots of good/bad writing, and see some good/bad trial advocacy.

Oh, and I would agree that neither is prestigious, but depending on what you want to do, these experiences could be looked upon very favorably.

OP needs to get on that rocket docket.