This will probably get moved to Basic Drug Discussion, but, since you know the amount in there and you know that it's 20mg/1mL you should be able to measure it out that way...with another dropper or another device that can do liquid measurement.
2mL = 40mg
1mL = 20mg
0.5mL = 10mg, etc
A teaspoon is 5ml.
Dry it out and use a scale, or dilute it.

you should have one of those measuring cups(mentioned above) that come with cough medicine somewhere in your house.. everyone has one (i thought) it would be safer to use that to measure a half tsp than actually filling a teaspoon half full.
but i understand desperate times, desperate measures (pun not intended)
Morphine isn't very active orally, half a tsp comes to 2.5mL, or 2.5 *20mg = 50mg morphine which may be a bit steep, depending on your prior opiate tolerances.
Have you ever used morphine before?
heres an idea.. take 1 tsp(100mg) and add to a known amount of water say using a measuring cup measure 1/2 cup water.. that would be around 100ml so now you know that 1mg/ml so by taking 1 tsp you would only be taking 5mg at a time.. much safer in my eyes.
Wait till they fall asleep or go to work.... or you know just go and grab one casually.

That sounds confusing but i think i'm getting it. Now I just need a measuring cup and this would be perfect (unless maybe you can't mix liquid morphine with water?).