Install 64-bit kernel in 32-bit Ubuntu

I want to use an application that is only available in 64-bit, but my Ubuntu installation is 32-bit. The application is proprietary, so I can't even recompile it for 32-bit.

I've read that I should be able to run a 64-bit application in 32-bit Ubuntu, if my kernel is 64-bit. From what I've read, this should work fairly smoothing with "MultiArch" support, which I should have in my version (14.04). But I can't figure out how to install the 64-bit kernel.

My processor seems to support 64-bit, because "lm" appears in the output of grep flags /proc/cpuinfo

I've seen a few questions already about upgrading the entire system to 64-bit, but those all tend to boil down to "just do a fresh install". I want to avoid doing a fresh install, if I can: I've got a lot of configuration I don't want to find and repeat, and I've read that on my old hardware with my relatively small amount of RAM, a full 64-bit system will probably use enough extra memory that it will get bogged down.

I've noticed that old kernel versions stick around until I remove them, so if I can install a 64-bit kernel, I assume I can boot into the 32-bit kernel again if things go wrong.

I found a guide to upgrading a Debian system that starts with installing the 64-bit kernel[1], but the instructions therein don't work for me:

$ sudo dpkg --add-architecture amd64$ sudo apt-get update$ sudo apt-get install linux-image-amd64:amd64

The first two commands seem to complete OK, but the install command gives:

Reading package lists... DoneBuilding dependency tree       Reading state information... DoneE: Unable to locate package linux-image-amd64

Am I doing something wrong? Missing a step? Should I just do a full install after all?

$ lsb_release -aNo LSB modules are available.Distributor ID: UbuntuDescription:    Ubuntu 14.04.1 LTSRelease:        14.04Codename:       trusty

[1] http://www.ewan.cc/?q=node/90