If I can offer you some advice, Nakedfingers, your husband and you need to be on the same page about this. If you rant and rave at him, he will withdraw, lie, do whatever it is that makes you stop. It's not the way to take away your fears about this friendship.
Regarding the ignoring in the supermarket, it sounds as if you and your husband have seen this friend there several times. Perhaps it's not the time/place and perhaps your husband has told her to keep her distance because you don't like the idea of the friendship. There could be a good reason for the distance in public places and it's not always bad news.
Presumably your husband loves you; you've already said that you love him so - have you talked to him about this? Calmly so that he can/will respond to you in the same way without being defensive? He shouldn't lie/misdirect you about this friendship but if you are on tenterhooks about it and liable to call his friend names, he will keep her from you -and she will keep her distance from you too.
Tell your husband how this friendship makes you feel and present a united front. If there's nothing going on then there's no harm done - and if there is, it will wrong-foot the friend. He's your husband but because he wants to be, not because he has to be and ditto your position as his wife.
Be calm until you know there is something to not be calm about.