Central Austin, that is.
The region outlined in blue below is what I usually mean by "central Austin." It's roughly the area afflicted by the original McMansion ordinance. (The map was prepared by the city demographer, Ryan Robinson, but the blue outline is mine.)
It's a little awkward to label tracts bordering Highway 183 and Capital of Texas as "central," but that's the convention. And although it might seem that this definition of "central" is so expansive it covers most of Austin, that is not true. "Central" Austin had 308,826 residents on April 1, 2010, just 39% of Austin's total population of 790,390, and only 18% of the metropolitan area population of 1.716 million.
"Not quite stagnant:" Austin added 133,828 residents between 2000 and 2010 for 20+% growth, but central Austin added only 7,672 residents, or about 2.5%. Central Austin's standard density increased from 4,311 ppsm to 4,421 ppsm.1
"On average": Most (47 of 76) tracts lost population.1 Most of these did not lose much, at least -- only two lost more than a thousand. Tract 23.11 (now split into tracts 23.17 and 23.18) gained the most; downtown (tract 11) gained the second most in absolute population but had the largest percentage increase, more than doubling.
Central Austin's weighted density increased from 6,117 ppsm to 6,512 ppsm.3 This was not an inevitable consequence of the (slight) rise in population. Rather, a handful of quite large, dense tracts experienced strong growth. In fact, if the two tracts in the University Neighborhood Overlay had merely maintained their population rather than growing like gangbusters, central Austin's weighted density would have dropped from 6,117 ppsm to 5,972 ppsm, even though total population still would have grown by 1.3%.
1 There are slight, but consistent, discrepancies between the Bureau's reported land areas for the 2000 and 2010 tracts despite no change in the boundaries. The difference may be due to a change in software, but I'm not sure. I used the land areas reported in 2010 for all calculations.
2 Four of the 2000 tracts (2.01, 3.01, 3.03 and 23.11) were split into new tracts for the 2010 Census (so there were 80 tracts in the 2010 census). To maintain consistency with the 2000 data, I recombined the split tracts.
3 The Census Burea still has not released the urbanized area data. If it does so before the next census, I'll re-calculate the weighted densities of large American urbanized areas. Based on the above, the weighted density of Austin's urbanized area probably dropped.