well in basement

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I have a similar problem. In my basement corner there is a 3ft. square x 4 ft. deep hole that is lined with stone. A well that is filled with water. It is always full, not really bothersome except when it rains heavy or snow melts. Then it overflows and has flooded the basement up to 3 inches deep. I have a sump pump now that keeps it down, but it too runs every few minutes, as the well fills back up.
I have drained the hole to see the bottom, there is a flow of water that steadily flows from underground to fill it up.It takes less than 2 hours to refill.
I'd like to cap it or seal it or control this water in some way to keep my floor relatively dry, or at least keep it from flooding.
This well is not my water source, there is a high water table here. Would it be as easy as driving a pipe down into the source, filling the wellhole with concrete, and capping the pipe somewhere above floor level?
What type of professional would be the best to contact?
Thanks.