Bermuda Grass has roots that go 6 to 8 ft into the ground...that is why it is so drought tolerant.

Vegetable garden or flower garden??

What color is the soil and what is the sun exposure, does the barn drain onto that area when it rains?? If the barn does drain on the area does it flood? Can you grade it so it does not flood?

You can spray the area with Round Up and burn it after the grass is dead, which will stunt the bermuda for a bit..But the bermuda will come back..

You can cover it with plastic (with mulch on top) and then cut holes for the plants you want to plant. (after you till it) The bermuda may still come up through the holes in the plastic. You will have to wait for a few weeks to plant after you spray the round up.

If you are planning a vegetable garden then you should til it and amend it. What you amend it with will depend on the aeration of the soil and color of the soil. Does the soil feel soft when you walk on it?

I have red clay so I amend the soil with horse manure (have horses) white oak leaves, sometimes grass clippings, sometimes peat moss, and every fall I amend it with cotton hulls (for aeration). This is your first year to plant in that area. If the ground is red clay (HARD/not aerated) you will need to do extra good amends on the soil.

Should I kill the intended area with RoundUp or some other non-select herbicide?
Yes. Several applications may be needed while the bermuda is actively growing for RU to be absorbed by the roots.

Do you recommend a soil test?
Yes.

I am concerned about regrowth of the grass, so what kind of mat or cover works best?

I would use the no- till method.

You may want to grow some veggies in containers while you prepare the garden.

Here is a link that might be useful: Soil, Compost and Mulch Forum FAQ

Check with the county extension office as they would be familiar with the soil in your area.
I agree with Scandia on the tilling and amending. The first year will be the most work. They have some new round-up that is extra strong and kills more. I'd spray the area and even out some to kill anything with runners on a day where no rain is expected for a couple of days. I would then rake and spray a second time with no rain forcast about 2 weeks after the first spray.
I did this with an area to make a lily bed. With living n the country I needed that area clear. I then amended with tilling using compost and manure. Never use fresh manure, it has to be completely dried out or buy the bagged manure as I did.
Good Luck
Lynn

Right; I forgot to mention not to use fresh manure..Do not use fresh saw dust either...Fresh Manure is acidic...much of the acidic tendencies are dried up with age..

Additionally, the cotton hulls take the winter to decompose properly so do not add cotton hulls to your garden until after the growing season.

Since you live in So Alabama I am assuming that your soil may be sandy..Adding compost and dried manure will help with this. Grass does well under some of the most adverse conditions...So that your grass is doing well, is not a good indicator that veggies/flowers will do the same..The grass maybe doing well because the barn drains on the area and feeds the grass with lots of water..

Watch the area the next time you have a chance...time the sun exposure...it is very important to observe whether the area floods during rain too, a nice level garden is better since this reduces nutrient run off. You do not want all your amends running off the garden with the first big rain.......Does the area get east to west sun?? Or is this exposure partially blocked by the barn?? Dig up some soil and put it in your hand. Come back here and let us know what the texture of the soil is..Is it sandy?? And what color is it??

I like to cover my garden with weed block or plastic with cedar mulch on top..(except where the plants are) So I do not have to clear weeds..Plus the cedar mulch deters insects.

Try lasagna gardening. If I have a really tough spot, I will use cardboard instead of newspaper. It has always worked for me.

Deb