Website Hacking Repair or Possibly More (See Description)

I'm looking to have two websites fixed and possibly have them rebuilt on a new server if deemed necessary.

My two company websites have fallen victim to repeated hacking over the last couple months. I've worked with a few people, some of which stated that there may be malware planted in the server that is allowing it to happen over and over again.

Our business is being affected at this point, so I need to get this solved immediately.

What I would ideally like is to avoid having to change everything and figure out once and for all what's going on with the sites, however I'm open to making the larger changes if absolutely necessary.

Budget is unknown and will depend on what's necessary.