bearpaws,
You know anything about keepin' bees? First rule is: don't fall and break your neck! Yer not gona get a lot of honey unless you get a chainsaw and open up that tree (and unless you got some experience, I wouldn't recommend doing that).
Next best thing is jest reach in there and pull out a perfectly cut piece of comb honey like the 'bee charmer' girl did in the movie, "Fried Green Tomatoes".
Get ya a bee veil (and probably some gloves too), tape up your pant legs and try to break off pieces of honeycomb toward the bottom (lower entrance area you saw) - that's where the honey will be. Put 'em in a clean plastic container and then run like he_-! Now is the right time of year since the bees have the most honey stored now but they'll also be pretty protective of it so be careful. If you work slowly and don't crush any bees when you rob them, they shouldn't be to difficult. Don't worry about how much honey to take...you're not gona get that much anyway without opening up the tree (most of it will be out of your reach).