How can I connect two networks across the Internet using the same subnet?

I need to be able to set up a remote office where the two networks are shared over the Internet. I need to have both the primary and remote offices on the same network subnet, due to some special software restrictions. VPN solutions won't work for me as they create a separate subnet. Remote access programs like Logmein require two sets of computers, so that won't work for us either.

I've searched the internet and found nothing but VPN solutions. It seams like there must be a way to restrict the two networks to a range of addresses and send network traffic over a secure Internet connection in such a way that neither office "knows" that they are connected over the Internet.

I should have specified that I need to connect two networks on the same LAN or internal subnet, separated from the Internet by routers at each end. It doesn't matter what the WAN addresses are, they are what the ISP offers. I'd rather avoid getting a dedicated T1 as small businesses can't afford the high cost. Thanks for your help so far, but I was hoping for an inexpensive solution.

Microsoft offers site to site VPN on their ISA product, but I'm not sure this will support having the two networks on the same LAN subnet, i.e Net A 192.168.1.1-100 Net B 192.168.1.101-200, connected over the internet so the connection between them acts just like a local connection.

This can't be done without a VPN or some other special point-to-point connection method, the cheapest of which is a VPN.

ISA can do what you want, as can anything that offers site-to-site VPN.

As for the IP Addresses: There's no reason why they both have to have the same subnet. You could have one on 192.168.1.x and the other on 192.168.2.x. This doesn't mean that your computers or users would know what's going on. If you browse the network you'll see all the computers in the domain, not broken up by subnets. In my office, at least, we connect to servers using DNS not IP address (e.g., you connect to "intranet" not 192.168.1.100).

set up a site to site vpn. there are a large number of smb vendors that sell this including cisco(most routers have options) microsofts ISA server, and even sonicwall has them

The Internet works because everything has a unique IP address.
Once you packets hit the Cloud they need to be directed to their destination. The Subnet is a major part of the IP address. The Subnet defines the number of Networks you can have per IP. Talk to your ISP about what you want to do; they can explain further why what you want isn't going to work. Also, talk to the software vendor about resolving the application problem.
Depending on what you might want to spend, some Cisco equipment can be set up to route data across a T1 Backbone. You could buy a T1 Point to Point, and route your network traffic across. Your application probably won't know what you are doing, but the Cisco Switches and Routers are going to run you $20k. The T1 Point to Point will cost $400 a month.
I suggest you consult Professional Telecommunications
contractor before you buy anything.