Composition instability

D5100, in (lets say) Aperture priority mode. I compose a photo by pointing the camera at a dark subject, get a corresponding low shutter speed. I press the shutter button half way down to focus. I then move the camera to point it at a lighter subject. I want it to retain the original low shutter speed, because I want it to be an over-exposed image. But it does not let me. It adjusts the shutter speed up, even though I am still holding the shutter button pressed half way down. Why is this, and how do I get it to stop?

Simples ! use AE lock button.
Half press focuses and takes a light reading. You can prob alter it in the menu, but I wouldnt.
Just use your AE lock. As long as you keep it pressed it locks the exposure, allowing you to refocus.

Brilliant. Thank you.

Your welcome
I will just add... Centre weighted metering is better for when using AE lock.
This will allow you more contol of where you place important exposure area !

Shoot in M mode. You control everything.

If you want to control shutter speed, would S mode not be the logical choice instead?