Its not the Network Adapter
First, thanks to everyone who responded to this problem. It hasn't been solved, but I will post it in the Networking Forum under the title, Connect Speed.
There was a lot of good advice given. Some of it I implemented, but didn't feel comfortable doing a flash bios unless I could go back to the original configuration.
It turns out that the motherboard was a PC Chips motherboard, and not an ECS as shown by Belarc. PC Chips and ECS may or may not be owned by each other or a single third party. The boards may be made at the same plant. However, this doesn't make them identical. ECS did not offer any help by saying it wasn't their board. They didn't mention PC Chips. PC Chips was helpful in pointing me to their flash bios sites, and they were responsive. I still don't have the answer on how to keep my current BIOS firmware if the new one doesn't work out. They didn't have my exact model number listed. They had M830L listed but not the LR model. The R stands for Riser; the phone modem is not built in, but put on a riser on the motherboard. So the M830L model was the one of interest.
Moved away from the problem by buying an ASUS A7V8X-X motherboard. The problem continued. Went to a reformatted hard drive with only the updated Win 98 SE OS, Zone Alarm and AVG installed. The appropriate drivers were also installed for the motherboard. Still slow connect speeds. Took the ethernet card from the old computer installed it on the new computer. No increase in speed. Bypassed the Router and went straight to the cable modem. The 8 year old Dell still downloads 5-7 times faster than the 2 1/2 year old one. The new computer downloads at 3-5 MB/sec while the old one downloads at 15-19 MB/s. A friend several blocks away gets 15-19 MB/sec on his computer also using Comcast
Thanks again for your responses, Georgexxx