According to the product page the lowest end 1060 EVGA sells needs 10A on the +12V rail (120W / 12V = 10A).  But you also need to consider the other parts of the computer.
 
Can you take a picture of the sticker on the side of the power supply and list what else is in the system (CPU HDD's etc.)?

In my last message I included a url but that got removed from the message so here is the info on the power supply I was thinking of going with, if the adapter approach would work:
 
Wattage: 400 Watt 
Type: TFX SFF 
Frequency: 60 / 50 Hz
Input Voltage: 115/230 Volts AC
Output Voltage: +5V 14A, +12V 22A, -12V 0.8A, +3.3V 13A, +5V-SB 2.5A 
Dimensions: ( W x H x D ) 3.4" x 2.5" x 7"
Connector included: ATX Connector 24 (20+4) Pin X 1 PC, P4-12V 4 Pin X 1 PC, SATA 15 Pin X 3PC, Molex 4 Pin X 3 PC, Floppy 4 Pin X 1 PC, PCI-Express 6 Pin x 1 pc
 

thanks. but how can the 6 pin PCIe only use 75W when EVGA says the card requires 120W?

regarding "If you get the power supply you listed, you don't even need the adapter (see text in bold)." - I mentioned above that "it would require an adapter since it only has 1 6 pin connector and the motherboard needs that connection"
regarding "The other 75W comes from the PCIe slot", are you sure of this? The 6 pin PCI-e connection on the Dell motherboard is used to power other things besides the expansion card slots such as HDD and DVD drives. 

cfmooney
regarding "If you get the power supply you listed, you don't even need the adapter (see text in bold)." - I mentioned above that "it would require an adapter since it only has 1 6 pin connector and the motherboard needs that connection"
regarding "The other 75W comes from the PCIe slot", are you sure of this? The 6 pin PCI-e connection on the Dell motherboard is used to power other things besides the expansion card slots such as HDD and DVD drives. 

Ah, good old non-standard OEM power supplies...  That 6-pin connector is not PCI-E, but a customized ATX pinout that eliminates other connections that the prepackaged system doesn't require to operate as advertised.  These kinds of prebuilts are typically not upgradeable because of this.  You probably won't be able to plug it into a video card anyway as the mold on the plug is different as to prevent people from trying to plug it in where you're not supposed to.  Doing so could cause shorts or even fires.  The power going to the PCI-E slots comes from this ATX power connection, but I don't know if the customized power layout in this PC is providing the full amount of power to the slot.
 
EDIT:  Reading comprehension fail...  You should be fine with a 2x Molex to 1x 6-pin PCI-E Y-adapter.  The drives that the PCI-E to the motherboard is powering should not be using more than 2A each.  Find out how much power the CPU is using and add the amperage from the other devices to find out if you have enough left over for the video card.