Can a wormhole supported by only small amounts of exotic matter really be traversable?

Recent studies have shown that (a) quantum effects may be sufficient to support a wormhole throat and (b) the total amount of ``exotic matter'' can be made arbitrarily small. Unfortunately, using only small amounts of exotic matter may result in a wormhole that flares out too slowly to be traversable in a reasonable length of time. Combined with the Ford-Roman constraints, the wormhole may also come close to having an event horizon at the throat. This Brief Report examines a model that overcomes these difficulties, while satisfying the usual traversability conditions. This model also confirms that the total amount of exotic matter can indeed be made arbitrarily small.