There are a couple of ways to accomplish this, and the level of complexity required varies a bit:
Option 1 - Serve on multiple subdomains
If the client is happy to have two subdomains for the two different parts of the company, you could have something like:
Serves pages for the primary company, potentially along with the "standard" about, contact, etc. type pages.
Serves pages for the secondary company, most of the navigation takes you back to www.example.com, but other interlinked pages are still held on othersite.example.com
This should be fairly easy to set up, but does expose the fact that there are two separate areas at work - however this may be sensible if you're requiring separate logins for the users/visitors.
As others have mentioned you may also want to add redirects on both sites to ensure that if people attempt to access a page that should be hosted on the other site they end up on the correct site.
This also allows both sites to easily have versions of the same page (for example if you want separate contact or login forms on both sections).
You can configure subdomains to point to different IP addresses, so these can be separate hosting providers or just different plans with the same provider.
Option 2 - Use a load balancer to split traffic
This is could be a much more complex option, but allows your client to serve both parts of the site with a single domain (i.e. www.example.com) but then internally route the requests to multiple servers.
The simplest way to then set this up would be for the secondary site to live under a directory on the main site (i.e. www.example.com/othersite), that way you can keep them separated, and have dedicated pages if you need them, but easily share pages if required. You would also be able to easily separate the media and other assets used by the pages.
You would then have a single published domain, but under the hood you would have two sites serving the application, and again these can be hosted with different providers as required.
Considerations with both options
Depending on traffic levels and the hosting plans you've chosen visitors may experience a slight lag when moving between sites if the sites need to be "warmed up" to start serving traffic.
You'll also need to duplicate a reasonable amount of templates, image assets, etc. between the two sites, and keep these updated if any of the site furniture, navigation items, etc. change. Links between the two sites will need to be fully qualified, however the DNS lookup required for the domains shouldn't really be noticable when moving between the two sites.