Shouldn't be too hard of a project, actually. What I would do, if it were mine, is remove as many tall or perennial weeds as possible, fertilize the bed well, lay down a thick mulch to kill off the sod (this won't hurt the asparagus), and then next year, fill in the gaps between the old plants with fresh roots, so that you end up with a nice, solid bed in a few years. If any grass were to try to come up through the mulch, I'd hit it with some Ortho Grass-B-Gone, which is perfectly safe to spray on Asparagus.

Your compost should be ideal for use as fertilizer and mulch.

Thanks for your response and time to answer. Your thought is my thought, except they won't allow any herbicides on the plot, which I suppose only means I need to weed the grass more vigorously. The plot was mowed about a month ago and in the last week many sprouts of new asparagus are coming up that haven't been there in decades. Pretty impressive plant it is. I'm marking them all with stakes and will mulch and fertilize around them. Later, I will add grass clippings. Any further thoughts you might have would be most welcome. Thanks again.

This is a reply to a really old link but others may benefit... I would dig up the whole bed in the fall and replant it. Do this when the 'ferns' start to turn a bit brown. There will also likely be lots of extra crowns to transplant to other locations, as there will not be room in the current bed to put them. At that time, remove the sod, and dig out the soil to asparagus planting depth, and sift it to remove as many roots as possible. Then plant the asparagus as recommended for your area, but mulch with lots of leaves (or other organic material like old hay or straw - not herbicide applied grass clippings - as some herbicides could harm the asparagus). Put on six inches or even more. It will compact overwinter and asparagus comes up through heavy mulch while grass does not. Next year you should have more asparagus that ever. Keep vigilant and weed out every blade of grass. I use POAST - a grass herbicide myself! But I replant too.