The tachyon mode in the open string spectrum is an indication that as a perturbation theory it describes the perturbation about an unstable vacuum. In 1999 Ashoke Sen realized that -- since the open string propagates with its endpoints on the space-filling D25-brane -- that instability must be the instability of the D25, which wants to decay to a "true bosonic string vacuum", usually called now the tachyon vacuum.
This conjecture became famous as "Sen's conjecture". It was subsequently checked to be true by numerical means in open bosonic string field theory. Moreover, using string field theory it became possible to study that non-perturbative state in which the D25-brane is indeed gone. There were several arguments that indeed in the vicinity of this state the perturbative open string has disappeared and turned into the closed bosonic string.
A breakthrough happened in 2005, when Martin Schnabl found an analytic expression for this "tachyon vacuum" in arXiv:hep-th/0511286.
For more references see here.