There are many methods to remove a unibrow. Plucking or waxing your eyebrows will not permanently remove the hair, but it will keep the hair away for a good month. Plucked/waxed hair does cause hair grow in finer.
To really permanently remove the unibrow, you'd need to get laser hair removal. It doesn't always take the first time, but the hair will grow in finer. In time, laser hair removal can be effective. There are products out there that claim pulse light therapy at home, but I'm not sure I'd use it on my face/eyes.
A uni-brow? well some say plucking makes is so it does not grow back as fast, girls wax it, I myself just use a razor for when it starts to look a bit Uni-brow-ish. you can always use Nar, but i dont know any permanent ways that dont harm the skin.
With laser hair removal. It is very cheap too. A lot of cosmetic surgery clinics do free consultation so you can just go and ask. It will remove the black hair for sure. The hair can keep growing as thin and blond. Laser is only a problem if you have darker skin. The completely permanent solution is skin replacement surgery. It is of course pricey. But it is hundred percent effective and permanent. They can take a piece of hairless skin out of anywhere else from your body and replace it with the skin that the uni-brow comes out. It will heal in a relatively short term. There is no known side effects of neither operation.