I wonder if it is a trailing Campanula? They tend toward blue lavender more than purple, though. Is it a dark purple flower and what shape?

how long does it bloom?
phlox subulata does that, bur you get about 2 weeks of massive bloom an then green the rest of the year.
vinca has dark green glossy leaves and blooms intermittantly
all the time.
diggerb

I'm guessing creeping plox, too. Lovely stuff, but short bloom time. I found that it suffers in too much direct sun.

How big is the box? How high? I'm thinking you could tuck in some annual trailing stuff, like sweet potatoe vine, intended for hanging baskets and window boxes. These plants tend to like a lot of sun.

Sounds like your describing an annual, my guess would be lobelia, you can buy the seeds any where, there purple and they trail, and bloom all summer.........mark

There are a number of annuals that meet that description. The mini wave petunias may be the one you are looking for.

Check out a good garden centre. Often the hanging baskets for sale have suitable plants that are also for sale as bedding plants.