What’s the best method for stopping leaking concrete slab?

There are only two things that really work:

1. Excavating the exterior and placing/replacing foundation drainage that will prevent water pressure from building up and pushing up through the slab. There are a number of ways to do this well.
2. Cutting a trench around the perimiter of the slab inside the building, filling the trench with a drain medium, and removing the water with a sump pump.

If your house is in a swamp, the latter may be the only solution. The first solution, however, is the most functional in the long term, as it may be accomplished without having to maintain a pump and the pump needing battery backup etc.