Best mulch to get for slimbed tapering toward path

Hi All,
I have some box that will take a few more years to do much shading yet. The weeds are already ridiculous and I am struggling to find time to tackle this part of the garden regularly, so I'd like to put something on top of the dirt to help me out a bit in preventing the weeds. The bed isn't very wide (maybe 1 - 2 foot).
The 'bed' slightly tapers toward a public path so I want one that's least likely to 'spill off''. I also don't wish to break the bank too much if possible? Would bark chipping work?
I have another much wider bed that I might do similar on so may be best to match, so thats where cost comes in a bit more...
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Gravel or bark are probably the best options Adam. Strangely enough, gravel is often a lot cheaper than bark - especially if you get bog standard pea gravel. Bark gets dislodged more easily so it may not be so good next to the path, but it looks nicer unless you go for a better quality gravel, so it really comes down to your preferences and pocket!
If you don't want to grow anything else on the area you could put landscape fabric/membrane down first before putting the mulch on. It suppresses weed growth from underneath and will save you having to weed so much as you'll only get ones seeding in from above. Just make sure to leave a little gap round your existing plants if you use it, and ensure the plants aren't short of water first. DIY places stock rolls of membrane, but garden centres usually do it by the metre so that might be cheaper depending on how much you need.
I live in west central Scotland - not where that photo is...