Finding that stone was meant to be.  Otherwise you would not have seen it.  They say that when the time comes, a crystal will leave you.  Maybe since your necklace broke, it is telling you that your use for it is over and you need to carry the amethyst.
Your life is changing and creating new experiences.  Go with it.  Let yourself feel its vibrations after cleaning away old ones. Hold it in your hand when you meditate to create a bond with it.  Everything has a vibration.  You just have to let yourself feel it.  You already know that it was meant to be yours, now enjoy it.

There is much good information in these responses.

One belief that I have, however, is that we can never really "own" a crystal or stone. Like a stray cat, they may come into our lives for a while, but then they may leave again. Crystals existed long before we were born and will continue to exist after we die. They have very long lives compared to us.

Amethyst is a very powerful and spiritual crystal. It is a good protector, because it transforms negative energy into positive energy.

Enjoy the amethyst while it is visiting with you! I'm sure it has much to share that you may benefit from.

right on everyone ...

one other thought i had to add ... your first instinct was to find the person it belonged to ... perhaps that's you, but perhaps it was asking you to explore your neighborhood and perhaps find a kindred crystal spirit right next door ...

i don't know what your layout is like, but if i found a stone on my driveway ... if it was one of my neighbors, there's only 4 neighbors (although that would be highly unlikely as everyone just gets into their cars in their garages and drives away) ... but perhaps it could have belonged one of the couple of kids who walk down my block on their way to school ... it would be easy to ask those people about it, and after exhausting that, i would know that it belonged to me, that it came to me.

if its a situation where there's a TON of people, e.g., you live in a highrise apt building in NYC or something ... pay attention when you are outside and watch and see if someone seems to be looking for something ... if so, asked if they have lost something ... perhaps this is the way the universe is trying to bring you together with someone who is important in the next step of your life ... it certainly wouldn't be the first time an amethest did something like that!!!!

but, like the other's said, until such time as it is clear it is not your's, you are the steward of the stone ... receive its energy and share your's with it as well.

peace & harmony,
elaine

Stones have a mind of their own and go to the person who needs them most.  As for feeling frantic when you lose a stone dont worry.  Stones will vanish when they feel they can no longer help you.  If it is ment for you to have the stone they will apear in your life again.  Hope that helps

It is also said a stone will find you, and if it is having this much of an impact on your life if personal "feel" it is to remain with you.
cleansing and recharging may not hurt either