I have cats, which handles the problem most of the time, although it does lead to the awkward problem of discovering a 'gift' at your doorstep when you go to let the dog out.
You could try humane traps. That way your dog won't get hurt - if the mouse won't get hurt, the dog won't, and I'm pretty sure you'll be able to reuse the trap in case of another mouse. Laying down poison, you run the risk of your dog getting into it. (Dogs are, I've found, almost as bad as cats when it comes to getting into things.) With sticky traps, well, your dog could end up getting his snout stuck to it.
You could try humane traps. That way your dog won't get hurt - if the mouse won't get hurt, the dog won't, and I'm pretty sure you'll be able to reuse the trap in case of another mouse. Laying down poison, you run the risk of your dog getting into it. (Dogs are, I've found, almost as bad as cats when it comes to getting into things.) With sticky traps, well, your dog could end up getting his snout stuck to it.