I am having problems connecting to the Minecraft server from a 2nd machine in the local network

The kids have got into Minecraft and want to run their own multiplayer game locally, however, we can't get one computer to connect to the other, but the reverse connection works.

COMPUTER A is the better machine and we want to run the server on that as well as a client. COMPUTER B can only really run either the server or the client. It's good enough to run TF2 etc. so it's not underpowered.

What works:

Run the server on COMPUTER B. We can run the client on COMPUTER A and connect to the server without any problems. We can also run the client on COMPUTER B and connect to the server, but it's unplayable.

What doesn't work:

Run the server on COMPUTER A. Running the client on COMPUTER A works and gives a decent frame rate etc. However, from COMPUTER B when we try to connect it appears to work, but then 30 seconds later it times out. The server reports that the connection has been made and then lost.

We have Sygate personal firewall on COMPUTER A which has a 1GB network card. This computer is also running IPv6.

We have Comodo firewall pro on COMPUTER B which has a 100MB network card. This computer is still running IPv4.

The two are connected via a 1GB switch.

Both firewalls have rules set up to allow all local traffic of any protocol on any port through. We haven't tried turning the firewalls off (that's for this afternoon), but I thought I'd see if there was anything else I could try.

UPDATE

The messages in the server are:

[INFO] User [/192.168.2.6:3222] logged in

then 30 seconds or so later:

[INFO] User lost connection: End of stream

We've just tried with firewalls on both computers disabled and got the following:

[INFO] Disconnecting /192.168.2.6:3172: Took too long to log in

Still at complete loss, but it does appear to indicate that it's a network connectivity issue.

UPDATE 2

I should have added that we're running XP SP3 on both machines. COMPUTER A is new hardware and (obviously) a new installation of all drivers and software. COMPUTER B is old hardware (hence the problem) but has recently been reformatted/reinstalled.

UPDATE 3

Just tried again with server version 0.2.4 and the latest client and we get the following message on the server log:

[WARNING] Can't keep up! Did the system time change, or is the server overloaded?

This is with nothing else running on this machine and no one connected. I would have thought an Intel Core 2 Duo E7500 @ 2.93GHz with 2 Gb of RAM and Physical Address Extension should be able to run the server software OK.

On Service Pack 3, there seems to be a lot of problems you apparently can't get around.

Quoting this topic: http://www.minecraftforum.net/viewtopic.php?f=1013&t=22790#p412298

[INFO] Player1 lost connection: End of stream
Researching
Problem:
  You're running Windows XP Service Pack 3.
Solution:
  Run Minecraft on a Virtual Machine on your computer,
  or use one with a different OS. Sorry!

Yo, I was having the same problem. I found a way. The problem was caused by Atheros L1 network adapter "number of receive buffers" count. The default is 256; after I increased it to 1024, the problem disappeared. This picture shows you how.

Update your network adapter driver on the host computer. Worked for me. I'm going to have a stab and also say that you are using XP SP3 on the host computer. Common problem, apparently...

A quick guess on this is the fact that you may be using your static ip address on client 2. If client 1 and client 2 are on a router, find the Dynamic IP of the computer that hosts the server. and try connecting with that. Also port forward if you haven't.