The second clause of the 17th Amendment gives the Governor the authority to, like an empty House seat "issue writs of election" to fill an empty Senate seat, however it goes on to say: Provided, That the legislature of any State may empower the executive thereof to make temporary appointments until the people fill the vacancies by election as the legislature may direct.
Thus, states have the authority to decide whether a Senate seat should be filled by appointment until the seat would have come up for election in the normal course of events, or fill the seat with a temporary appointment until an election can be held or some variant on that theme.
When Hillary Clinton resigned from the U.S. Senate to become Secretary of State, her seat was temporarily filled by the appointment by the Governor of U.S. Rep. Kirsten Gillibrand in 2009.
But, under New York law, that appointment was only good until the next regularly scheduled election which was in 2010. Hillary Clinton's term was scheduled to end in 2013, so Gillibrand has to run again in 2012 for a full six-year term.
One last point about offices without Members: Weiner's employees will maintain their jobs but will work for the office of the Clerk of the House as they respond to constituent requests and inquiries. The phones will no longer be answered "Congressman Weiner's Office" but will be answered with something like "New York 9th District office."
As of last night his official webpage was still available, but that will likely change by the end of the day.
Minority Leader Nancy Pelosi will decide who will fill Weiner's seat on the Energy and Commerce Committee.
See how this all works?