Welcome to the forums, McCollum!
A hydrometer is a valuable tool. But it would be pointless if you had to throw the must to use it. You can let it go inside the bucket without problem (just make sure its clean!) as long as you can retrieve it later (if you cant well no problem for the mead there, but maybe for you since you'll be screwed). Alternatively put some must in a tall vessel and use the hydrometer there, and then returning the mead to the bucket/carboy. Make sure the glass or wherever you put the must is clean, and when you are done make sure to wash the hydrometer, dry it well and store it again (the less you touch it with you hands the better. i wash mine with water and dry it with a paper without touching it directly, not too complicated, and that way i dont have to clean it again when im going to use it). Be careful since they break easily.
As rule of thumb here, dont put anything dirty in your must (or your must in anything dirty!), nor anything that was in contact with dirty things. As long as your hydrometer and company are well washed you are fine. Also consider buying a 100ml probete and a funnel. the probetes are high enough for most hydrometers and the low volume makes them easy to fill, and if you buy plastic, cleaning them is extremely easy, and they are cheap, and wont break the hydrometer. But thats just a personal recommendation.
Another person recently had problems using his hydrometer, and people gave great answers on how one works, so in case you are interested, here it is
http://www.gotmead.com/forum/showthread.php/26605-Sad-and-Embarrassing-Fact-I-Must-Face
Hope this helped