I tried this, and it doesn't achieve the effect I need.

The image is not purely just black and white. The edges are anti-aliased and so there is a slight gradient at the edge, with some pixels in various shades of grey. Your technique takes out the ones that are pure white but leaves the grey ones grey. I need those grey pixels to be partially transparent. eg for a pixel that is 50% grey in the original, I need it to be 100% black but 50% transparent. That way, if I superimpose the line art over a background colour that is not white, there's no grey "halo" round the edge. If I raise the tolerance higher, it takes out some of those grey pixels altogether but I don;t want them removed, I need them to be still there but "faded", so the background colour will show through.

Does that make sense?