Yes, Cane Creek makes a headset bottom for exactly this problem. I've used it on my MTB and it has worked just fine for over year. It does require that the original frame use zero stack or interior headset. But it looks like yours does.
Note: in the general case with a true 1-1/8" ( i.e. 34mm interior diameter, exterior headset cups), there isn't enough interior space to use a tapered fork.
From the Cane Creek Headset FAQ.
Cane Creek was the first headset company to offer a tapered solution for 1-1/8" head-tubes. Called XX-44, this bottom assembly enabled the use of 1-1/8" to 1.5" tapered forks on straight, 44mm head-tubes. This gave consumers more options, and frame designers more flexibility when creating the perfect ride.
Due to the popularity of the design, the EC44/40 (External Cup for 44mm head-tube, 40mm crown race seat) Bottom is now available in both the class-leading 40-Series and the premium 110-Series of headsets. This tapered solution fits 44mm head-tubes only; a "traditional" 1-1/8", 34mm version is not available.
Please note: Nominal insertion bore depth for 44mm head-tubes is 9mm. EC44/40 requires an insertion depth of 15mm. Park Tool Reamer 788 should be used with the supplied spacer to reach the 15mm bore depth for EC44/40 Bottoms.