Concrete strips are the way to go, a little bit of work but not much really. Very durable. Here's where i did a little retainer wall and used some concrete to make a clean line to mow. There's a length of re-bar along the length of the strip to keep it together, then just fill her up with concrete. This one strip probably used 2 wheelbarrows full of concrete. There's some colouring in the concrete.

* * * *

The secret is the soil.

Yeah Dianna, just dig a narrow strip. It'll only be as strong as you make it, so if you can, go a good few inches deep. 4" would be a good minimum but if it isn't holding anything back, you could go less.. Same for width, you want the strip wide enough to get the wheels of the mower over so the blades are over the concrete.
Lay a length of rebar in it, suspended off the bottom with whatever you have so it's in the middle, not touching the ground underneath.
The amount of soil you remove will be a good indication of how much concrete you need to mix. One wheelbarrow will go a fair way so you may want to do a small pour first to gauge the volume, or work it out as a rectangular prism,
L x W x H.
You could do a 20' job in an afternoon if you have all the stuff you need. Don't waste that topsoil and grass you remove..

* * * *

The secret is the soil.