The execution, however, was not broadcast by Aljazeera.
The group said it would kill the second captive within 24 hours unless its demands were met.
On 9 July, the group claimed it had captured the two men in Iraq and said it would execute them unless US-led occupation forces in Iraq released all Iraqi detainees.
The deadline was extended numerous times as Bulgarian authorities attempted to negotiate the captives' release.
The wives of the two men, identified by Bulgarian authorities as civilian truck drivers Ivailo Kepov and Georgi Lazov, made several videotaped appeals to the captors through Aljazeera news broadcasts.
Bulgarian Foreign Minister Solomon Passy made clear earlier in the week that Bulgaria's staunchly pro-US policy would not change as a result of the drivers' capture.
"Bulgaria is a stable state with a predictable foreign policy and we cannot expect it would change its foreign policy because of one or another group," Passy told state radio.
The company which employed the men in Iraq had on 9 July announced that it was suspending operations in Iraq.
Officials added on Wednesday that it does not have the power to influence an ultimatum from captors in Iraq to kill a second Bulgarian hostage.
This was "not in the power of the Bulgarian state," government spokesman Dimitar Tsonev said on state radio, after one hostage was confirmed executed.
He said it was up to "Iraqi authorities, given that sovereignty" is now in their hands, Tsonev said.
"They have the role to decide to free someone or no," Tsonev said.
Countdown for Egyptian
Meanwhile, another group thought to be the National Islamic Resistance has threatened to kill its Egyptian captive, Muhammad al-Gharabawi, within 72 hours unless the Saudi company he worked for pulls out of Iraq.
Al-Gharabawi, in his 50s, was abducted while delivering petrol products to US forces in Iraq by a group that it said represented the "legitimate Iraqi resistance", Aljazeera reported last week.
Faisal al-Nahait, who owns the transport firm where al-Gharabawi had worked for eight years, said unidentified callers had telephoned him several times since Wednesday demanding money.
He urged Egyptian authorities to help secure the driver's freedom.