Can a cardboard box be a 6 week long brooder?

Yes, I have done that before and done just what you said, cut one end out of each of two boxes and taped them together.  Worked perfectly. 

done it as well.   works great!

That's what I did, I got boxes from work (free), laid them on their longest/widest sides, cut the tops and duct taped it together. The final result is 4'x2' for our 10 chickies, and we only had to buy the duct tape.. Maybe, you could go to sears or another appliance store and see if they have any refridgerator boxes and I'm sure that would be big enough...

I got a box from Walmart. it's an Oscar Mayor Fully Cooked Bacon PDQ box. It's about 3ft x 3ft x 3ft

I don't have any chicks in it yet, so I don't know how long it'll last for them. But it's a great box cause it comes with a bottom.

If you look at post # 5 on this page: http://www.backyardchickens.com/t/634896/brooder-set-up-pictures-please

you can see what Ive done. Granted its for nearly 30 birds.. but this seems to be working really well. Just make sure that you put down at least one layer (preferably 2) of heavy duity tarp under them. That keeps spilage, leaks, ext from getting to the floor and makes clean up A LOT easier!

Go to a place that sells appliances and ask for a box.  If you have time in advance you can ask several places as they may not have any immediately.  My husband works for Home Depot and when I was brooding ducklings inside he brought me a refrigerator box.  It was GREAT and huge.  I blocked it off and started moving the divider as they grew.  It was wonderful for 7 ducklings for almost 2 months.  You can also get plastic drop clothes for the bottom.  I found it easier (at least with ducklings) to just use the cardboard as a surround and tape drop cloth to the bottom with shavings on top.  Ducklings are WAY more messy than chicks though so maintanence would be easier either way.