Mulch does help (the deeper the better), and there are some pre-emergent products which stop seeds from germinating. I remember there is even a mulch which is supposed to stop sprouting seeds (something with corn in it?) In my opinion, you have to check the flower bed about every week and get any weeds, no matter how small. I sometimes use a loop hoe to lift them, don't always pick up.

What kind of weed is growing there? If they are sprouts, it sounds like the plants are being left in place to drop new seeds every year. If you pull the weeds one year before they drop another batch of seeds, you should see exponentially fewer sprouts the following year.

Bark mulch is not very successful for weed suppression. Finely shredded wood works much better for that. It is still necessary to pull whatever unwanted sprouts show up though, with any kind of mulch. There is no kind of mulch that can completely prevent unwanted plants/sprouts.

If your weeds are perennials, they won't go away until pulled, dug, or smothered.