Things to consider:
1. A T1 is just a pipe to somewhere. It could be to the internet, it could be to another specific site. Make sure you know what is out there.
2. Depending on costs involved, you may find that your own 1.5MB/s DSL circuit costs less than sharing someone else's T1.
3. The customer buying that T1 service probably has a contract with the telcom vendor prohibiting them from reselling the service or bandwidth. If the other organization is a different legal entity, you may have a contractual problem. Your accounting department should also be consulted to make sure that if you do share the T1, they can properly allocate costs.
4. Domain permissions are a completely separate issue from T1 access. You will probably want a new router keeping your traffic separate from their traffic.
5. You should evaluate how much bandwidth you use now, and how much bandwidth the other organization uses now. If the total exceeds the available bandwidth for any long periods, this may be a bad idea.
Good luck.