In second area of Doppler's fortress, the glass in the background should be intact if Vile was destroyed (otherwise it will be broken). Make it past the area filled with water and, before entering the sub-boss gate, switch to Zero.

Enter the sub-boss room and defeat the sub-boss. Make sure the final blow is dealt with Zero's Saber. (Just fully charge up, shoot it twice, wait for it to come down, then dodge and slash it.) It will slowly move up then go kamikaze on Zero.
X will then teleport in after the whole exploding sequence, and Zero will hand him the Z-Saber. To use it, just use it like when Zero was using it: Charge up to green, fire two Buster shots (still combinable), then push the attack button for the slash.

Note: With the Buster Upgrade, X can launch a sonicboom with the Z-Saber, which, if landed on an enemy that doesn't get instantly defeated, will deal extra damage on the enemy for up to 4 times. One single shockwave can take off approximately 2/3 of a boss' life, INCLUDING ALL versions of Sigma, provided that the hit did land.