What is probably happening here is you have power to the light on one black, another black to the switch, and a return path along the whites. That is, the switch is AFTER the light.
This is generally unsafe as the light is still energized, even when the switch is off, and should be fixed regardless.
The first thing I'd do is beg/borrow/steel a non-contact voltage detector.
With everything still attached, switch the light off. If I'm right, the detector would read hot on BOTH black wires, and the whites will be un-energized. Flip the switch and the whites should be energized as well.
If this is indeed the situation, the fix is relatively easy. Turn the power off! Remove the existing light.
Figure out which pair is wired to the switch. The easiest way is to use a meter in resistance mode. Find the black-white pair that shows low resistance when the switch is on, and open circuit when switched off. Mark the white wire with a loop of electrical tape. This will be the switched hot.
Now wire the blacks together with an appropriate wire nut, and make sure it's a secure connection. You should not be able to see any copper.
Wire the switched hot (the white wire marked with tape) to the black wire of the new fixture. Wire the remaining white to the white wire of the fixture. Make sure you've grounded the fixture according to the manufacturers instructions.
Mount the fixture, and turn the power back on.
Portable lamps (at least in the US) are generally designed without a ground wire. The bodies of the lamps, even if metal, are designed not to be likely to be prone to a short to the hot lead. The standard lamp cord is therefore two wire (hot and neutral). Most modern lamps are polarized, meaning the cord/plug is designed to ensure that the hot lead is attached to the hot side of the outlet.
Sconces and other lamps that are intended for fixed mountings on electrical boxes have a hot lead, a neutral and a ground. The ground is the safety measure in case of a short.
When you convert a fixture intended for use with a ground wire to a plug-in setup, you need a grounded cord to preserve the safety factor. The simplest way may be to buy a grounded extension cord, cut off the socket end and splice the cord with the plug (male) end into your fixture.
Attach the hot lead to the black wire of the fixture, the neutral to the white (the neutral wire is attached to the larger blade of the plug and the casing is sometimes ridged), and the ground to the green (or bare).
You can then run the cord under a cord cover to a conventional grounded outlet.
P.S. Be sure to use a bushing or something similar to prevent any cord/cable from being pulled through and cut or frayed by metal edges at the point where it exits the fixture or metal cable cover.
CAUTION
This whole discussion assumes that you are installing this permanently (a wall fixture is not well protected at the back from damage or short) and on an interior wall. Exterior fixtures are specifically designed to attach in an exact way to exterior boxes to ensure that they are water resistant. Any compromise of that attachment mechanism may lead to leakage, shorts and danger of electrocution or fire.
Best Answer
Sometimes these types of lights are used to indicate a status. For instance, if the bottom light is on, the patient wants a nurse and if the top light is on the nurse needs assistance. Other scenarios are that one would have a low wattage bulb used as a night light and the other would be a higher wattage for reading and what not.
You can wire them both together and run it to a single switch. That's perfectly okay.