Unfortunately, you have to have a cross-browser CSS (so, Solution 1 is not a quite a valid solution, you should not have accepted it formally, it won't work for different browsers; I'm not sure the claim in the last sentence is correct, but does work on latest Gecko (Mozilla), but it would not work on earlier versions).
The solution is well-known. I would advice to use one of the rounded-corner generators to obtain required CSS:
http://cssround.com/[
^].
You can refer to the layout engine comparison to see what features are supported by which:
http://en.wikipedia.org/wiki/Comparison_of_layout_engines_%28CSS%29[
^],
http://en.wikipedia.org/wiki/Comparison_of_layout_engines_%28HTML5%29[
^].
—SA