The elbo section of duct work in basement fell down from the duct it was connected to and previous owner had it connected with duct tape. New application of duct tape did not hold it for long. So what is the safest way to connect these two ducts or at least prop it up to the ceiling so it cannot drop out again?

You should use aluminum duck tape made for heating and air ,very sticky stuff will last forever wal-mart for about 5bucks

HVAC supply stores have the stuff necessary to hold ducts together. It's very sticky, high temperatuure and has a removable backer material.

See: Duct Tape - Good for Many Things, but NOT for Ducts (http://www.consumerenergycenter.org/myths/duct_tape.html)

Example:

3M US: Scotch HVAC Duct Tape (http://www.3m.com/product/information/Scotch-HVAC-Duct-Tape.html)

Use terms Duct tape HVAC for Google