I would make the curve flat by not bending a mesh to fit it, but "beveling" the curve (adding thickness).
- Add a BezierCircle (in addition to the curve you already have)
- In edit mode convert the handle type to Vector (super quick way V,V), that will make it a square.
- Rotate it 45° along the Z axis (just to make it straight along an axis).
- Scale the square along the X axis, to make it a thin ribbon.
- (Last step) On your other curve object set the Bevel Object to be the "BezierCircle" you added in step one.
Place an object so its origin point is located in the same position as a curve's origin point.
Subdivide the object as many times you like (being in Edit Mode press Ctrl+R and scroll the mouse wheel to increase the number of cuts or just type the number from your numerical keyboard).
Add a Curve Modifier to the object. In a window under Object: type the name of your curve.
Enable Display modifier in Edit Mode option to easy adjust the lenght of an object and make it fit the curve perfectly. It is very handy as you may change the object's parameters and see how it looks placed on a curve in the same time.
To follow up with images from Mr Zak's comment, which specified the answer: