$ sudo gem install dm-mysql-adapter --no-rdoc --no-ri
Fetching: data_objects-0.10.3.gem (100%)
Fetching: do_mysql-0.10.3.gem (100%)
Building native extensions.  This could take a while...
ERROR:  Error installing dm-mysql-adapter:
	ERROR: Failed to build gem native extension.
/opt/ruby-1.9.1/bin/ruby extconf.rb
checking for mysql_query() in -lmysqlclient... no
*** extconf.rb failed ***
Could not create Makefile due to some reason, probably lack of
necessary libraries and/or headers.  Check the mkmf.log file for more
details.  You may need configuration options.
[...]

This means while compiling the extension, compiler can’t find mysql librairies. It’s because they haven’t been installed in standard include folders. They actually are in /usr/local/mysql.
To indicate to the compilation process to check this folder you need to add the following parameter to the command line:

$ sudo gem install dm-mysql-adapter --no-rdoc --no-ri -- --with-mysql-dir=/usr/local/mysql/

Building native extensions.  This could take a while...
Fetching: dm-do-adapter-1.0.2.gem (100%)
Fetching: dm-mysql-adapter-1.0.2.gem (100%)
Successfully installed do_mysql-0.10.3
Successfully installed dm-do-adapter-1.0.2
Successfully installed dm-mysql-adapter-1.0.2
3 gems installed

You’re done!

Example:

$ gem install --no-rdoc --no-ri rails