This is an excellent question to consider. My short answer to it is that he does tell the truth partly, but clearly he does not reveal the way that he had rather arrogantly hoped to be the man responsible for healing the breach between the two feuding houses of the Montagues and the Capulets. This question of course does relate to a much bigger issue, which is to what extent is Friar Lawrence responsible for the tragedy.
However, to focus on this speech alone, on the one hand, Friar Lawrence openly confesses everything about what he had done and his own role in the tragic events. He admits that he was the one to marry Romeo and Juliet and that also he gave Juliet a sleeping potion and hoped to manage the reuniting of the lovers. He does end his speech by saying that if he can be found at fault, then he accepts any punishment that the state does give him.
However, if we look more closely at what he says, we could argue that he is trying to present himself in a way that lessens his guilt in the case. Consider the following example:
You, to remove that siege of grief from her,
Betrothed and would have married her perforce
To County Paris. Then comes she to me
And with wild looks bid me devise some mean
To rid her of this second marriage,
Or in my cell there would she kill herself.
Note the way that in this quote Friar Lawrence implicates the Capulets in the tragedy, accusing them with the rather stark "You." He also presents himself as being forced into giving Juliet the potion to stop her from killing herself, whereas, although Juliet certainly was suicidal and desperate, we could argue Friar Lawrence was driven by thoughts of the glory that he could achieve by broaching a peace between the two houses. So, the Friar is certainly truthful on the whole, but we need to be aware of the spin that he puts on his version of events and how this spin deflects blame.