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Here's the relevant statute:

§3-10-3. Vacancies in offices of state officials, United States senators and judges.
Any vacancy occurring in the office of secretary of state, auditor, treasurer, attorney general, commissioner of agriculture, United States senator, judge of the supreme court of appeals or in any office created or made elective to be filled by the voters of the entire state, judge of a circuit court or judge of a family court is filled by the governor of the state by appointment. If the unexpired term of a judge of the supreme court of appeals, a judge of the circuit court or judge of a family court is for less than two years or if the unexpired term of any other office named in this section is for a period of less than two years and six months, the appointment to fill the vacancy is for the unexpired term. If the unexpired term of any office is for a longer period than above specified, the appointment is until a successor to the office has timely filed a certificate of candidacy, has been nominated at the primary election next following such timely filing and has thereafter been elected and qualified to fill the unexpired term. Proclamation of any election to fill an unexpired term is made by the governor of the state and, in the case of an office to be filled by the voters of the entire state, must be published prior to the election as a Class II-0 legal advertisement in compliance with the provisions of article three, chapter fifty-nine of this code and the publication area for the publication is each county of the state. If the election is to fill a vacancy in the office of judge of a circuit court or judge of a family court, the proclamation must be published prior to the election as a Class II-0 legal advertisement in compliance with the provisions of article three, chapter fifty-nine of this code and the publication area for such publication is each county in the judicial or family court circuit.

From Wikipedia

He was sworn in on January 4, 2007, to his ninth term; the term will expire on January 3, 2013.

If the Governor names a replacement before July 3, 2010, then his replacement must go through the primary process and stand for election. If the Governor waits until after July 3, his successor finishes out the term.

Any guesses on how long it will take the Dem Governor to find a replacement?

If the Governor names a replacement before July 3, 2010, then his replacement must go through the primary process and stand for election. If the Governor waits until after July 3, his successor finishes out the term.

So Byrd is not dead until Joe Manchin says he's dead?

For ease of reading, Yossarian reformatted it thus:

Here's a reformatted version, redone in pseudo-code style:

WEST VIRGINIA CODE
§3-10-3. Vacancies in offices of state officials, United States senators and judges.

Any vacancy
   occurring in the office of
      secretary of state,
      auditor,
      treasurer,
      attorney general,
      commissioner of agriculture,
      United States senator,
      judge of the supreme court of appeals
   or
       in any office
          created or made elective
            to be filled by
               the voters of the entire state,
               judge of a circuit court or
               judge of a family court
is filled by
   the governor of the state by appointment.

If
   the unexpired term of
      a judge of the supreme court of appeals,
      a judge of the circuit court or
      judge of a family court
   is for
      less than two years
or if
   the unexpired term of
      any other office named in this section
   is for a period of
      less than two years and six months,
the appointment to fill the vacancy is for
   the unexpired term.

If
   the unexpired term of
      any office
   is for
      a longer period than above specified,
the appointment is until
   a successor to the office
      has timely filed a certificate of candidacy,
      has been nominated at
         the primary election next following such timely filing
      and
      has thereafter been elected and qualified to fill the unexpired term.

Proclamation of
   any election to fill an unexpired term
is made by
   the governor of the state
and,
   in the case of
      an office to be filled by the voters of the entire state,
   must be
      published prior to the election as a
         Class II-0 legal advertisement
            in compliance with
               the provisions of article three, chapter fifty-nine of this code
            and
            the publication area for the publication is
               each county of the state.

If
   the election is to fill a vacancy in the office of
      judge of a circuit court
      or
      judge of a family court,
the proclamation must be
   published
      prior to the election as a
         Class II-0 legal advertisement
      in compliance with the provisions of
         article three, chapter fifty-nine of this code
      and
      the publication area for such publication is
         each county in the judicial or family court circuit.

Note: Code updated with legislation passed through the 2009 4th Extraordinary Session

I read it as “the unexpired term” of the candidate who is no longer in office or, as in this case, dead. So, how does it matter when an appointment is made, if the candidate died more than 2 years and 6 months before the end of the unexpired term?

Here comes some more Dim nuance. Does the vacancy occur upon Byrd's death (pre-July 3rd) or upon Gov Manchin declaring the vacancy (surely post-July 3rd)?

The devil in the details will be the definition of when the vacancy occurred.

In addition to Gov Manchin, Secy of State Natalie Tennant is also a Dem.