Carefully snip a slit on the inside of the waistband near the side seams, making sure you don't go through to the right side of waistband. Repeat at the opposite end.
Thread the elastic through to the opposite slit.
*make sure the elastic is the width of the waistband*
Machine the first slit closed by machining to and fro a few times across the waistband making sure you stitch through the elastic.
(If stitching by hand make sure you stitch through the elastic and use a strong button thread.)
Pull the other end of the elastic to the width of the child's waist making sure it is a comfortable fit.
Repeat the machining of the other slit.