Rings that fit on a Picatinney rail will also fit on a weaver base. The dimensions are the same, except for the spacing of the cross-slots. On a Weaver, they are usually only on slot is for each ring to be mounted, whereas the Picatinney has slots that are spaces about .5" apart regularly spaced all the way down the rail. ARMS makes rings like that. QD, strong, holds zero, though pricy.
BTW, if you have one scope that you are swapping between a Bolt gun and an AR, the rings needed for them are different heights. ARs typically need high rings to allow proper cheek weld whereas Bolt guns need lower type rings. Also the scope will need to be zeroed everytime its placed from the AR to the bolt, and vice versa. This would be a serious pain in the ass. My suggestion is to save up for another scope altogether.
Ghost