If we were to (baselessly) assume that the "week" spoken of here was indeed Daniel's seventieth week, it still would not even imply what you imagine, that is, that the week continues immediately after the sixty-ninth. For if, instead of only quoting past of the passage at hand, we examine the entire passage, we find:
"The Epistle of Barnabas"
Chapter 16
Moreover, I will also tell you concerning the temple, how the wretched [Jews], wandering in error, trusted not in God Himself, but in the temple, as being the house of God. For almost after the manner of the Gentiles they worshipped Him in the temple. But learn how the Lord speaks, when abolishing it: “Who hath meted out heaven with a span, and the earth with his palm? Have not I?” “Thus saith the Lord, Heaven is My throne, and the earth My footstool: what kind of house will ye build to Me, or what is the place of My rest?” Ye perceive that their hope is vain. Moreover, He again says, “Behold, they who have cast down this temple, even they shall build it up again.” It has so happened. For through their going to war, it was destroyed by their enemies; and now: they, as the servants of their enemies, shall rebuild it. Again, it was revealed that the city and the temple and the people of Israel were to be given up. For the Scripture saith, “And it shall come to pass in the last days, that the Lord will deliver up the sheep of His pasture, and their sheep-fold and tower, to destruction.” And it so happened as the Lord had spoken. Let us inquire, then, if there still is a temple of God. There is — where He himself declared He would make and finish it. For it is written, “And it shall come to pass, when the week is completed, the temple of God shall be built in glory in the name of the Lord.” I find, therefore, that a temple does exist. Learn, then, how it shall be built in the name of the Lord. Before we believed in God, the habitation of our heart was corrupt and weak, as being indeed like a temple made with hands. For it was full of idolatry, and was a habitation of demons, through our doing such things as were opposed to [the will of] God. But it shall be built, observe ye, in the name of the Lord, in order that the temple of the Lord may be built in glory. How? Learn [as follows]. Having received the forgiveness of sins, and placed our trust in the name of the Lord, we have become new creatures, formed again from the beginning. Wherefore in our habitation God truly dwells in us. How? His word of faith; His calling of promise; the wisdom of the statutes; the commands of the doctrine; He himself prophesying in us; He himself dwelling in us; opening to us who were enslaved by death the doors of the temple, that is, the mouth; and by giving us repentance introduced us into the incorruptible temple. He then, who wishes to be saved, looks not to man, but to Him who dwelleth in him, and speaketh in him, amazed at never having either heard him utter such words with his mouth, nor himself having ever desired to hear them. This is the spiritual temple built for the Lord.
Early Church Fathers - – Ante-Nicene Fathers: The Writings of the Fathers Down To A.D. 325.
When we examine all this together, we plainly see that whoever wrote this miserable document was clearly teaching that the new temple he was describing was to have been built AFTER the old temple and the people had been destroyed in a war. Now we know that the war he was speaking of did not take place within the seventy weeks following the going forth of the order to restore and to build Jerusalem. Some claim that the seventy weeks were completed before our Lord suffered at Calvary, but most say that only sixty-nine weeks had been completed at that time. But either way, approximately 41 years later, when the city and the temple were destroyed, was clearly not within the entire span of the seventy weeks. So, even if the writer of this document meant the seventieth week, he was clearly not saying that this week flowed continuously from the sixty-ninth week.
There is simply no way to logically argue that this confused document even implies that the seventieth week of Daniel took place immediately after the sixty-ninth week.