The same problem appears to have occurred during the upgrade from 3.8.0-19 to 3.8.0-21. In other words, this bug appears to be of broader scope than just the release upgrade.

I worked around the problem by deleting the abi, initrd.img and System.map files for 3.8.0-19. I did so with much trepidation, because other life priorities have reduced the time I am able to spend to keep my Linux knowledge as sharp as it once was.

This is the screen output from the upgrade process: 

Running depmod.
update-initramfs: deferring update (hook will be called later)
Examining /etc/kernel/postinst.d.
run-parts: executing /etc/kernel/postinst.d/apt-auto-removal 3.8.0-21-generic /boot/vmlinuz-3.8.0-21-generic
run-parts: executing /etc/kernel/postinst.d/initramfs-tools 3.8.0-21-generic /boot/vmlinuz-3.8.0-21-generic
update-initramfs: Generating /boot/initrd.img-3.8.0-21-generic
run-parts: executing /etc/kernel/postinst.d/pm-utils 3.8.0-21-generic /boot/vmlinuz-3.8.0-21-generic
run-parts: executing /etc/kernel/postinst.d/update-notifier 3.8.0-21-generic /boot/vmlinuz-3.8.0-21-generic
run-parts: executing /etc/kernel/postinst.d/zz-update-grub 3.8.0-21-generic /boot/vmlinuz-3.8.0-21-generic
Generating grub.cfg ...
Found linux image: /boot/vmlinuz-3.8.0-21-generic
Found initrd image: /boot/initrd.img-3.8.0-21-generic
Found linux image: /boot/vmlinuz-3.8.0-19-generic
Found initrd image: /boot/initrd.img-3.8.0-19-generic
Found memtest86+ image: /memtest86+.bin
done
Setting up amarok-doc (2:2.7.0-0ubuntu2.1) ...
Setting up amarok-common (2:2.7.0-0ubuntu2.1) ...
Setting up amarok-utils (2:2.7.0-0ubuntu2.1) ...
Setting up amarok (2:2.7.0-0ubuntu2.1) ...
Setting up linux-image-extra-3.8.0-21-generic (3.8.0-21.32) ...
Running depmod.
update-initramfs: deferring update (hook will be called later)
Examining /etc/kernel/postinst.d.
run-parts: executing /etc/kernel/postinst.d/apt-auto-removal 3.8.0-21-generic /boot/vmlinuz-3.8.0-21-generic
run-parts: executing /etc/kernel/postinst.d/initramfs-tools 3.8.0-21-generic /boot/vmlinuz-3.8.0-21-generic
update-initramfs: Generating /boot/initrd.img-3.8.0-21-generic

gzip: stdout: No space left on device
E: mkinitramfs failure cpio 141 gzip 1
update-initramfs: failed for /boot/initrd.img-3.8.0-21-generic with 1.
run-parts: /etc/kernel/postinst.d/initramfs-tools exited with return code 1
Failed to process /etc/kernel/postinst.d at /var/lib/dpkg/info/linux-image-extra-3.8.0-21-generic.postinst line 1010.
dpkg: error processing linux-image-extra-3.8.0-21-generic (--configure):
 subprocess installed post-installation script returned error exit status 2
dpkg: dependency problems prevent configuration of linux-image-generic:
 linux-image-generic depends on linux-image-extra-3.8.0-21-generic; however:
  Package linux-image-extra-3.8.0-21-generic is not configured yet.

dpkg: error processing linux-image-generic (--configure):
 dependency problems - leaving unconfigured
Setting up linux-headers-3.8.0-21 (3.8.0-21.32) ...
No apport report written because MaxReports is reached already
                                                              No apport report written because MaxReports is reached already
                                                                                                                            Setting up linux-headers-3.8.0-21-generic (3.8.0-21.32) ...
Setting up linux-headers-generic (3.8.0.21.37) ...
dpkg: dependency problems prevent configuration of linux-generic:
 linux-generic depends on linux-image-generic (= 3.8.0.21.37); however:
  Package linux-image-generic is not configured yet.

dpkg: error processing linux-generic (--configure):
 dependency problems - leaving unconfigured
Setting up linux-libc-dev:amd64 (3.8.0-21.32) ...
No apport report written because MaxReports is reached already
                                                              Setting up libkscreen0 (0.0.92-0ubuntu0.1) ...
Setting up kscreen (0.0.92-0ubuntu0.2) ...
Processing triggers for libc-bin ...
ldconfig deferred processing now taking place
Errors were encountered while processing:
 linux-image-extra-3.8.0-21-generic
 linux-image-generic
 linux-generic
E: Sub-process /usr/bin/dpkg returned an error code (1)
A package failed to install.  Trying to recover:
Setting up linux-image-extra-3.8.0-21-generic (3.8.0-21.32) ...
Running depmod.
update-initramfs: deferring update (hook will be called later)
Examining /etc/kernel/postinst.d.
run-parts: executing /etc/kernel/postinst.d/apt-auto-removal 3.8.0-21-generic /boot/vmlinuz-3.8.0-21-generic
run-parts: executing /etc/kernel/postinst.d/initramfs-tools 3.8.0-21-generic /boot/vmlinuz-3.8.0-21-generic
update-initramfs: Generating /boot/initrd.img-3.8.0-21-generic

gzip: stdout: No space left on device
cpio: write error: Broken pipe
E: mkinitramfs failure cpio 1 gzip 1
update-initramfs: failed for /boot/initrd.img-3.8.0-21-generic with 1.
run-parts: /etc/kernel/postinst.d/initramfs-tools exited with return code 1
Failed to process /etc/kernel/postinst.d at /var/lib/dpkg/info/linux-image-extra-3.8.0-21-generic.postinst line 1010.
dpkg: error processing linux-image-extra-3.8.0-21-generic (--configure):
 subprocess installed post-installation script returned error exit status 2
dpkg: dependency problems prevent configuration of linux-image-generic:
 linux-image-generic depends on linux-image-extra-3.8.0-21-generic; however:
  Package linux-image-extra-3.8.0-21-generic is not configured yet.

dpkg: error processing linux-image-generic (--configure):
 dependency problems - leaving unconfigured
dpkg: dependency problems prevent configuration of linux-generic:
 linux-generic depends on linux-image-generic (= 3.8.0.21.37); however:
  Package linux-image-generic is not configured yet.

dpkg: error processing linux-generic (--configure):
 dependency problems - leaving unconfigured
Errors were encountered while processing:
 linux-image-extra-3.8.0-21-generic
 linux-image-generic
 linux-generic