Cable internet two computers phone line

Hi
I want to set up internet for a second computer in the house. The first computer is connected to a cable modem, in an upstairs room. The second computer will be downstairs.
What I have been planning is to get a router and then run an ethernet cable to the downstairs computer. My question however is this. There is already a regular phone line than runs from the upstairs room through the house, above the ceiling tiles in the downstairs room. The phone line is not used. Is there any way to run the internet connection through this phone line, or do I need to use a new ethernet cable?? I am thinking it is not likely, but it would save me some work and money. Thanks.

Not really, one twisted pair (a phone line) is insufficient for Ethernet, although many years back it could be used for Apple Talk in the pre-MacOS era.

Ethernet cable has 4 twisted pairs and uses two of the pairs. You need CAT 5e cable for the run from upstairs to downstairs or you could consider either wireless or Powerline (like Ethernet over AC wiring) if actually running the cable is too onerous.