How much time do you have? Another means, rather than using herbicides or digging them up, is to cut them close to the ground and put a tarp on them. This process is called occultation. It works much better with small weeds rather than ones with large root systems. But the principle is the same: with less exposure to light, the weeds will be weakened, and their growth above ground level will be limited, and their root systems will be weakened.
After performing this process for a long period of time (2-3 weeks at a time, 4-5 times), you can use a tiller to get deep into the root systems and break them apart. If you apply a sheet mulch like cardboard, using potentially 2-3 layers and apply your soil and compost on top of that, it will have three substantial benefits: it will inhibit the growth of the bramble, and it will act as a mulch, retaining moisture. Finally, it will also just break down into your soil and enrich it.