It's not easy to guage the width of your garden but i'll take a punt on 20' wide - if you have measurements it'd be helpful.
Are you prepared to put in a little more work?
Don't just slap a bed on top of the grass, as scbk says the plants need a depth of soil for roots and, as important, drainage. If you lay soil directly on top of grass the water will collect and leave the area sodden.
Are you willing to rearrange the path to leave a wider area for the flower bed, rather than lay it along the fence...and how far down are you going with it?
If you curve the path with an 'S' shape it will give a curve for the flower bed to fit in and provide an interesting meander through the area, or a series of --|--|-- slabs laid in a set of steps to give a bit of height; not much, maybe 6" or so. It all depends on your budget and willingness to use a bit of imagination before you begin the job. No matter how small a space it is, you can make the most of it.