First peel the rubber boot off the back bearing, now cut some narrow strips of paper tape (masking tape, painters tape, whatever). Wipe the grease off...
One problem with Dremel tools is that they don't always age gracefully, after a number of years of service they can develop a lot of lateral play in the spindle. Since the majority of uses for a Dremel place a side load on the spindle, it deforms the bearing mounts over time, I'll show you a quick trick to tighten them back up.
This Dremel I picked up at a garage sale had almost 1/16 of an inch of visible lateral play before.
Pretty simple, unscrew the annulus behind the collet and remove the four T15 screws. Pull apart the two case halves, remove the spindle/armature, and it's as apart as it needs to be.
Step 2: The problem
Here you can see the problem, the back bearing is inside a rubber mount, probably for vibration purposes, and the front bearing is held in a little webbed plastic housing, over time the rubber softens and the plastic deforms, we just need to slightly increase the diameter of our bearings to make every thing tight again.
Step 3: The fix
First peel the rubber boot off the back bearing, now cut some narrow strips of paper tape (masking tape, painters tape, whatever). Wipe the grease off the outside of the bearings, and wrap two to three turns around the bearing, replace the rubber boot and reassemble.
After doing this my Dremel now has no visible play in the spindle.