In version 10.1 it appears "FullData"
will do this, but it also produces some extra fluff that will need to be trimmed.
dat = Import[
"https://docs.google.com/spreadsheets/d/1F3uGfNJH-\
hch3NfnvZvqzg8vqmMLUkMF2dgAsHPP66s/gviz/tq?tqx=out:html&tq&gid=0", "FullData"];
dat[[2, 2 ;;, ;; 7]] // Grid // TeXForm
$\begin{array}{ccccccc}
341367901 & \text{} & 0 & 0 & 0 & \text{} & 0 \\
341367902 & \text{} & 0 & 0 & 0 & \text{} & 0 \\
341367903 & \text{} & 0 & 0 & 0 & \text{} & 0 \\
341367904 & \text{} & 0 & 0 & 0 & \text{} & 0 \\
341367905 & \text{} & 0 & 0 & 0 & \text{} & 0 \\
341367906 & \text{} & 0 & 0 & 0 & \text{} & 0 \\
341367907 & \text{} & 0 & 0 & 0 & \text{} & 0 \\
341367908 & 6.3 & 6.3 & 0 & 0 & \text{} & 0 \\
341367909 & 5.3 & 11.6 & 0 & 0 & \text{} & 0 \\
341367910 & 5 & 16.6 & 0 & 0 & \text{} & 0 \\
341367911 & 5.8 & 22.4 & 0 & 0 & \text{} & 0 \\
341367912 & 5.4 & 27.8 & 0 & 0 & \text{} & 0 \\
341367913 & 5.4 & 33.2 & 0 & 0 & 278 & 278 \\
341367914 & 5.3 & 38.5 & 0 & 0 & \text{} & 278 \\
341367915 & 6.2 & 44.7 & 0 & 0 & \text{} & 278 \\
341367916 & 1 & 45.7 & 0 & 0 & \text{} & 278 \\
341367917 & 4 & 49.7 & 0 & 0 & \text{} & 278 \\
341367918 & \text{} & 49.7 & 0 & 0 & \text{} & 278 \\
341367919 & 3.6 & 53.3 & 0 & 0 & 219 & 497 \\
341367920 & \text{} & 53.3 & 0 & 0 & \text{} & 497 \\
341367921 & \text{} & 53.3 & 0 & 0 & \text{} & 497 \\
341367922 & 3.9 & 57.2 & 0 & 0 & \text{} & 497 \\
341367923 & 7 & 64.2 & 0 & 0 & \text{} & 497 \\
341367924 & 5.9 & 70.1 & 0 & 0 & \text{} & 497 \\
341367925 & 5.6 & 75.7 & 0 & 0 & \text{} & 497 \\
341367926 & 5.9 & 81.6 & 0 & 0 & 260 & 757 \\
341367927 & \text{} & 81.6 & 0 & 0 & \text{} & 757 \\
\end{array}$