Note (10/02/2013): For information on installing Java 7 on the Beaglebone Black, see this post.
My new Beaglebone Black arrived yesterday. There isn't an OpenJDK 6 package for the version of Ångström Linux on the Beaglebone Black as there was for the previous Beaglebone. This means that you need to locate and download the .ipk files that opkg uses to install software.
You can install Java by following the instructions at http://beaglebone.cameon.net/home/doing-java-development. Use the links provided on the page to download the .ipk files needed to install the packages listed.
You may need to install libgif4 before you can install openjdk-6-common. If you do, be sure to run opkg update first, then try opkg install libgif4.
Before you can install the openjdk-6-vm-zero package, you will need to download and install the libffi5 package. You can download the libff5 package from http://www.angstrom-distribution.org/repo/?pkgname=libffi5. You want the armv7a version. The installation process is the same as described on the Cameon page at the link above.