There is no way for someone to find out you worked somewhere for 6 weeks.
Well, yes there is. I don't know why you think that would be hard, if you received a paycheck.
Sorry if you don't believe that, but it's true.
I have seen the background checks on every single person hired in my region since 2008, and per my companies parameters, we see at least the last 2 positions they were at, and exactly how long they were there.
If they worked somewhere for 1 week, and it was within the last 2 jobs they had, it will show up, and I will, and have, asked them about it. If we wanted, we could have the background check give us the dates on the last 3 jobs.
Most typically the response would be it was some temp assignment, or they were doing a short term seasonal job.
Never lie about your past work history. If you are found out, you will be branded a liar, with good cause, because you are.
If you explain the short tenure honestly and fairly, without blaming anyone, chances are you will be treated honestly and fairly. I have gone to bat for more than 1 person we hired, because I saw so much good in them as a future employee, and they were able to use the experience gained as a growing experience. Just about 6 weeks ago, I asked someone "what did you learn from that?" Instead up making up some complicated, convuluted answer, she laughed (softly) and said "I learned not to do THAT again!" I laughed too. That's exactly what I would have expected her to learn. She was hired, and is, so far doing very well.
Everyone makes mistakes. Don't lie about it.