On Wednesday, August 31, 2011 11:44:17 PM UTC-7, wrote:
Silver-soldered to be precise.

One of the bars of the stainless steel BBQ grill detached itself at
one end from the stringer...
I silver-soldered it back into place ... the joint... gave way.

Brazing/silver-soldering is best when it has large contact
area with the right (about .005") gap for the filler material.
Perhaps your round/round wire joint just doesn't have
the right area; a suitable workaround would be to clean
and flux, then wrap thin wire around the joint (like a lashing
with string). When you apply heat and solder, this will make
a brazed joint with all the area of the thin wire, and with the
wire acting as a reinforcing fiber.

You might want to peen the working surface flat, using a block
of wood to back the grid, afterward.