Although I quoted Meggido's 32-yr-old LP algorithm in the comments, a simpler-to-implement randomized algorithm has expected linear time and space complexity.
![EnclosingCirc][1]
(Image from [code,project.com](http://www.codeproject.com/Articles/22568/Computational-Geometry-C-and-Wykobi).)
A clear description of this algorithm may be found at this lecture prepared by Marc van Kreveld and Maarten Loffler: [PDF slides download](http://www.cs.uu.nl/docs/vakken/ga/slides4b.pdf).