Is there an inexpensive device that fits into a telephone jack into
which the phone and computer are connected which would allow telephone
service while connected to the internet without having to subscribe to
DSL or cable hookup?

I don't think so ... Wow what an idea!

Maybe one of those clever Researchers could invent one. Better still,
invent two ... I'm sure that I could sell the other.

Looking at how a traditional home dial up works, 

1) When you use DSL - your phone circuit is permanantly connected to
your internet provider at the local switch.  (either your local
carrier provides the dsl or leases your circuit to the dsl company). 
When you use dial up, you use your phone circuit to actually go across
the phoneline past the local switch to make a connection to your
isp.... This will not allow you to send and recieve calls while your
circuit is being used.

2) DSL is transmitted over non-audible frequencies- it can co-exist
with voice transmissions.  Traditional dial up uses audible
frequencies so they can not co-exist.

Since both sides of a dial up connection have to speak the same
"Language" anything you did at your side would have to be repeated at
the recieving end of a dial up connection also...

Since DSL is dropping in price - $26/month in Chicago with a 1 year
contract. I doubt much work is being done to improve traditional dial
up....
 
Just wait for wireless broadband where your laptop can use cell towers
to maintain a permanant broadband connection to the internet....
without any phone lines.

As vballguy-ga says, when you make a dial-up connection, your modem is
just making a normal telephone call. So if you want to "allow
telephone service while connected to the internet" via dial-up, you
have two alternatives:

1) Make two simultaneous telephone calls by (for example) getting a
second telephone line; or

2) Use the dial-up connection to get telephone service by (for
example) getting VOIP over dial-up.

If these alternatives are too costly for you, do you really need
simultaneous service? Perhaps all you need is notification of an
incoming call, with identification of who is calling you, and the
ability to either take the call or ignore the call.

That kind of arrangement can be provided as a service. A search for
"internet call waiting" will turn up several providers.

You can also get call waiting and caller ID and a gadget to place
between your modem, your telephone connection, and your telephone. The
gadget detects the call waiting signal, alerts you to the incoming
call, and displays the telephone number. (If you get caller ID with
name, it can also display the name.) If you pick up the attached
telephone, you are immediately connected to the caller. If you ignore
the call, your internet connection is unaffected.

Examples of these gadgets include the AT&T 438 Internet Call Alert,
the Command Communications Hot Call 4000, the ClassCo Internet Call
Waiting Model 5000, etc. If the attached telephone can display the
caller ID, you can choose simpler gadgets (such as the Catch-A-Call),
but make sure it passes on the caller ID (some do not.)

Call waiting and caller ID are included in many local telephone plans,
or are available as relatively inexpensive options.

SEARCHES:
VOIP over dial-up
internet call waiting

what u want is an ISDN services, ask to telcom company if they have any 4 u :)

ISDN requires TWO lines!

Better with ADSL.

What a cool idea.

Rural areas and many areas in the Far East (poor infrastructure) get
internet access via the Power Lines.  They bypass the phone and cable
lines altogether.

There's always "NetMeeting" by Microsoft