A pilot reported they descended below their assigned altitude and received a Low Altitude Alert from ATC.

Narrative

My mission was to take 3 passengers and cargo to ZZZ and then return to PAFB with one passenger and some cargo. This occurred on my return flight. I was on an IFR flight plan from ZZZ to PAFB; filed route ZZZZZ. My filed and cleared altitude was 8;000. I was in IMC and icing conditions and was using the aircraft's anti ice system to mitigate the icing. I requested a lower altitude from ATC if able. Shortly after my request the controller cleared me to 6;500. I repeated my cleared to descend altitude of 6;500. Once I said it; I reached up to the autopilot and sequenced the altitude bug to 6;000. My mouth said 6;500 but my brain and hand were thinking 6;000 and I had not noticed my error. I descended to 6;000. Shortly thereafter; ATC issued a Low Altitude alert to me with instructions to immediately climb to 6;500.I repeated my assigned altitude correctly but I sequenced it in the autopilot incorrectly and didn't catch my error. Situational awareness is key to flying; especially in IMC. In order to prevent this from occurring again; I will build a new mental/physical system for me to use: Once I say my cleared altitude back to ATC; I will sequence it and say it out loud again to myself to verify my mouth; my brain; and my hand are all on the same page. I understand the dangers of an altitude deviation; especially a low altitude alert from ATC. This was a very good lesson for me to learn.Human performance considerations that certainly contributed the the event that occurred: My stress and fatigue level was higher than normal due to several factors. I hadn't slept very well the night prior to the incident I had a mission the day prior and I didn't get back to PAFB until late. I met my 10 hours of rest; but I didn't sleep well during then. Not as experienced in IMC as I would like. Icing conditions building up on one wing more than the other; needing the anti ice system on high to de ice both wings properly.