There is no clear answer. In some old versions of the Arthurian legends, it appears that the sword in the stone is the sword that Arthur carries throughout his life (thus Excalibur or Caliburn, some of the many variations on the Welsh name Caledfwlch). In Malory, Arthur breaks his first sword and goes to the lady of the lake for a new one and that was Excalibur. In some of the old French versions, it is Gawaine who carries Excalibur, not Arthur. Modern versions tend to conflate the two swords (sword in the stone and gift of the lady of the lake). Some scholars have concluded that though they were two different swords, they had the same name.
My own opinion is that a) there would be no reason for the Lady of the Lake to give him a sword if he already had an excellent sword, and b) there is no reason for Arthur to send his sword that he carried through his life (Excalibur) to the Lady of the Lake when he was dying, unless he had received it of her in the first place. Therefore, the sword given to him by the Lady of the Lake is Excalibur.