Nope, only one connection will be able to access the Internet. If you have weak signal strength, you need to address that, not try to cobble something like this together. I'd start by changing the router's channel, and also you can add hi-gain antennas or signal boosters to one end or both.
Hawking Tech has a number of products that will help you increase your wireless range. The root page is
Hawking Hi-Gain™ WiFi Range Extending Products.
Some of the more interesting products are this
Hawking [HSB2] Hi-Gain WiFi Signal Booster, which can be used on either end of a wireless connection to boost the signal power.
Another way to increase your signal strength is by the use of hi-gain antennas. You can choose from omni-directional or directional models, here are a some examples.
Hawking [HAI7SIP] Hi-Gain 7dBi Omni-Directional Antenna Hawking [HAI15SC] Hi-Gain 15dBi Corner Antenna [HAO14SD] Outdoor Hi-Gain 14dBi Directional Antenna Kit
This is just a sample of available products, many people have hi-gain antennas with similar specifications, but I haven't seen any other suppliers of signal boosters.