If you cover any plant with something that keeps it from getting the sunlight that plant needs to grow it will die. For the most part I have found over the years that it takes between 6 to 8 weeks, although some things, Bishops Weed is one, can take longer becasue of the food reserves the plants store in the roots. Bishops Weed is tough and may take years to eliminate, and it may try growing other places. You may need a vertical barrier between where the Bishops Weed is and the Buxus hedge.
Covering with newspaper, or cardboard, would be a good start however.

Thanks for the answer kimmsr. Did you mean that it takes years to eliminate with cardboard and plastic sheet aproach?

Sarong, in my climate at least (mild, rainy, much like England) the conventional advice is that it can take a year of smothering before roots of bishops weed have rotted.

I did get it out of a smaller area (maybe 6x10'?) that I didn't want to wait to replant by digging, sorting through the soil to remove as much root as I could, then being very attentive to any little bits that were resprouting and pulling those sprouts immediately for the next couple of years.

You may want to deadhead any blooms so it can't set seed, leaving you with only the root spread to deal with in areas where you don't want to dig or aren't going to remove it.

It could take a year, or more, for the roots on many invasive plant species, such as Bishops Weed to die out. It can take that long even with "weed killers", especially with the plants that will shut off the flow of sap when the plant detects that a poison has been applied. One of the reason most of those "weed killers" do not work as well as the manufacturer attempts to make you believe so you spend money on their product and still have the "weed" problem.
Some "weeds" produce lots of seeds as well as grow as biennuals or perennials while others protect the roots, where energy is stored for new growth, until the plant finds the sunlight it needs to photosynthesize the nutrients the plant needs to grow. Eliminating a plants access to light, which the plant needs to grow, is the best way to kill it, as well as the least expensive method.

Ok, so I am looking at at least a year of having my garden covered. Thats ok I guess since its just too big to clean manually. I am planning to clean out smaller beds by hand, I already started but we got bellow 0C temperatures again so earth is frozen. Hm, would it work to till the land where it is and let it freeze to death?
There is just so much of it everywhere, and those are only the places I found so far. I fear once the spring comes I will see that there is more. Strange thing is that we are the only garden around here that has it... Well at least we will not get infested again from the neighbors..