Undriven element when connecting coax to ground?
A dipole should be driven with a balanced signal or it ceases to become a dipole - my bet is that you are missing some subtlety like a balun made from short pieces of coax. Measuring with a meter is a waste of time of course because, at DC ground will connect to one of the dipole's terminals because of the balun. Only at the proper RF frequency can this be measured.

Regards the comments about not feeding a dipole with a balanced signal, here's what I found below.

Wiki to the rescue. Look at the "Feeding a dipole antenna" section: -


  When a balanced antenna is fed with a single-ended line, common mode
  currents can cause the coax line to radiate in addition to the antenna
  itself,[11] distorting the radiation pattern and changing the
  impedance seen by the line. The dipole can be properly fed, and retain
  its expected characteristics, by using a balun in between the coaxial
  feedline and the antenna terminals. Connection of coax to a dipole
  antenna using a balun is described in greater detail below


And another rescuer: -


  In Figure 1, a coaxial cable is connected to a dipole antenna. For a
  dipole antenna to operate properly, the currents on both arms of the
  dipole should be equal in magnitude. When a coaxial cable is connected
  directly to a dipole antenna however, the currents will not
  neccessarily be equal. To see this, note that the current along a
  transmission line should be of equal magnitude on the inner and outer
  conductors, as is typically the case. Observe what happens when the
  coax is connected to the dipole. The current on the center conductor
  (the red/pink center core of the coax, labeled IA) has no where else
  to go, so must flow along the dipole arm that is connected to it.
  However, the current that travels along the inner side of the outer
  conductor (IB) has two options: it can travel down the dipole antenna,
  or down the reverse (outer) side of the outer conductor of the coaxial
  cable (labeled IC in Figure 1).


Figure 1 (as mentioned directly above): -