To make hot rolled steel strap from strap like cast starting material cast pieces can be first cut from the solidfying casting after the casting process is completed. Then these cast pieces are guided to and stored in an oven where they are brought to and maintained at the rolling temperature. The cast pieces are fed in succession into the rolling mill. Then the cast pieces are stored there at the rolling temperature until the rolling process begins. A number of cast pieces are produced as raw products in our invention and stored until the rolling of these individual cast pieces starts. the casting is continuous and occurs also during the rolling operation. Because of the intermediate storage provided by the oven significant differences can exist between the casting time and the rolling time. Assembly and maintenance of the rolling mill, e.g. roll changing, can be performed in a pause of the roll cycle which includes the rolling time. If necessary the two pauses of two roll cycles are placed immediately in succession. The cast pieces formed during this time are fed to the oven and stored there. An apparatus for performing this process is also described.