I received a small amount of assistance with this on StackExchange: http://gis.stackexchange.com/questions/63444/reprojecting-natural-earth-admin-0-countries-shapefile-to-web-mercator
Short answer: The 1:50m shapefile has its southernmost line at approximately -89.9989, while the 1:10m is at -90.0. When attempting to reproject the portion of the feature at -90.0, the projected point goes to infinity and triggers the error.
In order to work around the problem, I was able to trim off the bottom of Antarctica using the “Reshape Feature” tool in QGIS. There’s a decent description of how to use it here: http://gis.stackexchange.com/a/21651/1770. After removing the southernmost portion of Antarctica, I was able to successfully reproject the map.
To save everyone some time, I’ve posted the files. These are the “_lakes” files that do NOT include boundary lakes:
* Original file in WGS 84 (EPSG:4326), but with the bottom 0.4 degrees of Antarctica trimmed off. You should be able to reproject this file without issue: https://github.com/dmillz/misc/raw/master/shapefiles/ne_10m_admin_0_countries_lakes-trimmed_antarctica.zip
* Same file, but reprojected into Web Mercator (EPSG:3857): https://github.com/dmillz/misc/raw/master/shapefiles/ne_10m_admin_0_countries_lakes-EPSG_3857.zip
I hope this helps other folks who might run into the same issue :-)
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