We report a deep search for HI 21cm emission from the $z= 0.00632$ sub-DLA toward PG1216+069 with the Giant Metrewave Radio Telescope. No emission was detected and our $5\sigma$ upper limit on the mass of any associated galaxy is $M_{\rm HI} \lesssim 10^7 ~M_odot$, nearly 3 orders of magnitude less than $M_{\rm HI}^\star$. The $z \sim 0.006$ absorber is thus the most extreme known deviation from the standard paradigm in which high column density quasar absorption lines arise in the disks of gas-rich galaxies.