Remove the fitted sheet from the top of the mattress. Turn it inside out. Hold the fitted sheet by one corner.
Push the pointed end of a seam ripper under one of the threads holding the hemmed fabric edge over the elastic. Pull up on the seam ripper, tearing the thread.
Continue tearing the threads holding the fabric over the elastic until the elastic is exposed. Pull the elastic out of the hole you've made in the hem and cut it with scissors.
Pull the elastic out through the hole. Continue pulling until you have removed the elastic from all the way around the fitted sheet.
Sew the hole in the hem closed with a needle and thread that matches the sheet. Remove the mattress and pull the fitted sheet over the box spring. It will fit snugly, but since there is no elastic edge it will hang down instead of tucking under the box spring. Replace the mattress.