You could install concrete underpinning to extend the stone foundation walls to below the proposed basement elevation. The underpinning would be incrementally installed using a series of individual underpinning piers. You have to decide whether it would be easier to underpin from outside the building or to work inside with low headroom and have to carry the excavated spoils out of and the concrete into the basement.
Even though you underpin the basement walls, you still have to check that the stone walls do not slide off the top of the concrete piers. If the building is very lightweight, this could be a consideration. If heavy, less so.
The underpinning should be continuous around the perimeter of the building. The underpinning piers are usually (almost always) unreinforced concrete, no dowels, no non-shrink, no keyways. A space about 3 inches high is left between the top of the concrete underpinning and the bottom of the stone foundation. This space gets drypacked with a slightly moist, sand-cement mixture (about 3:1 ratio). After a pier is poured, it is drypacked the next morning. As soon as a pier is drypacked, an adjacent pier can be excavated. There is no need to wait for the concrete or drypack to cure. Waiting just adds significant time and cost to the project. 4000 psf dirt = 28 psi. Overnight, concrete develops sufficient strength to support the building.
The underpinning pit width must be limited to approximately 3 to 4 feet along the stone wall to prevent damage to the wall. If the stone wall will not temporarily span this distance, you may have to close the width or provide some temporary wall support. Particluar care must be used if there are any column footings or pilasters along the stone foundation walls. You don't want to undermine too much of a column footing or pilaster at one time. Also, you should not concurrently excavate underpinning pits within about 8 feet of each other (edge to edge) unless there are completed, concreted, drypacked piers between them. The sides of the underinning pits shoud be shored with 2 x 10 boards to assure proper pier shape, especially inside the building. For deeper pits, the shoring is necessary to protect the worker in the pit and to prevent excessive undermining of the foundation wall adjacen to the excavated pit.
Underpinning is usually too expensive for a homeowner. If so, you could deepen the basement away from the stone walls. Maybe just deepen a 25' x 35' area with a curb wall around the perimeter to retain the soils under the stone wall.
Also, before you start any excavation or underpinning on the building, you should perform a pre-construction survey and establish movement controls (horizontal and vertical). Even with the best underpinning procedures and an experienced underpinning contractor, it is not unusual to settle the building a little (plus or minus 1/4 inch). It is unrealistic to specify zero movement. Just excavating the overburden from along the bottom of the foundation wall (without even digging under the wall) is sometimes enough to cause a little settlement.
References: "Handbook of Temporary Structures In Construction" by Ratay and "Foundation Engineering Handbook" by Winterkorn & Fang
