A well string assembly for drilling a borehole in a subsurface formation, the assembly comprising: - an upper tubular well string part having upper and lower ends between which there is formed a passageway; - a lower well string part having upper and lower ends (10, 11) which lower end is connectable to or includes a drill bit; - a releasable well string interconnection means for selectively interconnecting the lower and upper well string parts (5, 8) above the lower end of the lower well string part; and - an auxiliary tool arranged such that it can be passed along the passageway of the upper well string part, wherein the auxiliary tool comprises an operating means for manipulating the well string interconnection means so as to disconnect the lower well string part from the upper well string part, and wherein the auxiliary tool is arranged such that it can be passed, after disconnecting the upper and lower well string parts (5, 8), through an opening at the lower end of the upper well string part, so as to reach a working position in which at least part of the auxiliary tool is not radially surrounded by a part of the well string, whereby the lower end of the lower well string part is spaced apart from the lower end of the upper well string part when the upper and lower well string parts (5, 8) are connected to each other, wherein disconnecting the lower well string part disconnects the drill bit from the upper well string part, characterised in that the auxiliary tool further comprises a lower well string connection means for connecting the auxiliary tool to the lower well string part simultaneously with or before disconnecting the well string interconnection means. - an upper tubular well string part having upper and lower ends between which there is formed a passageway; - a lower well string part having upper and lower ends (10, 11) which lower end is connectable to or includes a drill bit; - a releasable well string interconnection means for selectively interconnecting the lower and upper well string parts (5, 8) above the lower end of the lower well string part; and - an auxiliary tool arranged such that it can be passed along the passageway of the upper well string part, wherein the auxiliary tool comprises an operating means for manipulating the well string interconnection means so as to disconnect the lower well string part from the upper well string part, and wherein the auxiliary tool is arranged such that it can be passed, after disconnecting the upper and lower well string parts (5, 8), through an opening at the lower end of the upper well string part, so as to reach a working position in which at least part of the auxiliary tool is not radially surrounded by a part of the well string, whereby the lower end of the lower well string part is spaced apart from the lower end of the upper well string part when the upper and lower well string parts (5, 8) are connected to each other, wherein disconnecting the lower well string part disconnects the drill bit from the upper well string part, characterised in that the auxiliary tool further comprises a lower well string connection means for connecting the auxiliary tool to the lower well string part simultaneously with or before disconnecting the well string interconnection means.