I there a stock digital camera that will take the old Olympus OM1 lenses?

Would save me a ton of $$$ of there was.

There's a lot of adapters (quite cheap) for most of the digital cameras with exchangeable lenses (mirrorless or dDLT
E.g. For Micro 4/3 (Olympus and Panasonic cameras), Sony M-alpha mount (Nex, some alpha models), Canon EOS and certainly also M-EOS cameras, Fuji FX, Nikon V and still some more.
Those adapters have in common that they don't need any glass in them as they only have to make up for the longer lens to sensor distance the OM lenses need. So you only need an adapter that puts the lens further away from the camera body. Easy to make.

The ones available from good online shops should be a bit more precise than the 10$ adapters from China but also many of the China ones are OK.

Nevertheless you have some limitations when using your manual focus lenses on a digital body.
1) no autofocus (of course)
2) works only in M and aperture priority (you have to set the aperture, the camera calculates the exposure time)
3) Focusing can be tricky so make sure the camera has a true optical viefinder or an electronic viewfinder. Focusing using the camera screen is really a pain especially when shooting handheld you easily find yourself one hand short.
4) Autofocus confirmation: Some camera bodies need a signal from the lens telling htem the lens is focused before you can release the shutter. With most cameras this can be turned off if that is not possible you need an adapter with an electronic chip. This chip tells the camera that the lens is focused, no matter if it really is. It only tricks the camera.

I guess that's the main topics when using a manual focus lens (like your OM lenses) on a digital body.

The adapters for OM lenses are quite cheap, as I said, so it they could save you some good money, especially in the beginning. Nevertheless you might consider buying a good AF lens for your digital system later on.

Here is an example of a OM lens to Micro 4/3 camera adapter:
http://www.amazon.com/...002U4SBG4/

I hope this helps and don't forget to chose a best anwer, it motivates people to keep answering questions. There's an adapter to mount OM lenses onto Micro Four Thirds cameras--mirrorless Olympus or Panasonic system cameras.
http://www.bhphotovideo.com/...apter.html" class='ext_link'>http://www.bhphotovideo.com/...apter.html
If one of those is the camera that you want (possibly a very good choice), then you might as well try it. I doubt that it will save you money, as you get tired of no autofocus or metering and have to buy Micro Four Thirds lenses anyway.
If it is not the kind of camera that you want, don't let your choice be a slave to your old lenses.
There's no adapter that can mount your OM lenses on a DSLR. OM-System lenses and lens accessories can be used on Olympus E-System DSLR and PEN digital cameras with an adapter.

Totally manual operation of course… No.

Olympus changed their mount when they went digital. You can use old Olympus film lenses using an adapter with mirrorless cameras. They should work in both manual and aperture-priority modes. No AF but the lightmeter should work.

There's an adapter to mount OM lenses onto Micro Four Thirds cameras--mirrorless Olympus or Panasonic system cameras.
http://www.bhphotovideo.com/c/product/631816-REG/Olympus_260051_MF_2_OM_Lens_Adapter.html
If one of those is the camera that you want (possibly a very good choice), then you might as well try it. I doubt that it will save you money, as you get tired of no autofocus or metering and have to buy Micro Four Thirds lenses anyway.
If it is not the kind of camera that you want, don't let your choice be a slave to your old lenses.
There's no adapter that can mount your OM lenses on a DSLR.

OM-System lenses and lens accessories can be used on Olympus E-System DSLR and PEN digital cameras with an adapter.

Totally manual operation of course…

No.

Olympus changed their mount when they went digital. You can use old Olympus film lenses using an adapter with mirrorless cameras. They should work in both manual and aperture-priority modes. No AF but the lightmeter should work.