Just in case you have nodata value in your original raster, SAGA Constant Grid would come in handy.
From QGIS Processing toolbox, go to
SAGA | Raster Tools.
On the Constant grid window:
Name: Give any name (
my_raster, ...) (*1)
Constant Value: Your aimed value.
Data Type: Select your required data type (*2)
Target Grid System: Select
 grid or grid system
Target System [optional]: Select your original raster layer.
(*1) Honestly I do not know why this is needed.
(*2) as of QGIS 2.18.19, it seems SAGA returns Float32 raster regardless which
Data Type we choose. If you need integer raster (e.g. 0-255), please try
Convert data storage type to obtain
1-byte integer following this operation.
The appearance of output layer (name
Target Grid) may not look successful, in the Layers panel.
If in doubt, please check the cell values by (1) Load min/max values. or (2) using
Identify Feature tool.