In Act III, Scene 1 of Romeo and Juliet, Romeo returns from his marriage to Juliet by Friar Laurence to find his friends.  But, he sees an altercation in progress and seeks to defuse the situation.  Now that he is married to Juliet, he tells Tybalt that he loves him and does not wish to quarrel.  Calling upon Benvolio to get Tybalt and Mercutio to lay down their swords after the enraged Tybalt draws his sword, he stands in the way of Mercutio, causing him to be stabbed.  After the mortally wounded Mercutio is taken off, Romeo bemoans his weakness:

...O sweet Juliet,

Thy beauty hath made me effeninate,

And in my temper softened valor's steel (3.1.115)

Here Romeo refers to his sword, that he uses against Tybalt, just as Tybalt has slain Mercutio with his sword. 

It is only logical that Romeo would use the same type of weapon that Tybalt does.  Also, the sword is the weapon that the men of that time used.  For instance, in the first act, Sampson, the Capulet servant tells Abraham, "My naked weapon is out," an allusion to his sword which has been pulled from its sheath.  Later, old Lord Capulet calls for his "long sword" when he hears the feuding.  So, with the mention of "steel" for his weapon, as well as the other mentions of swords, it seems very likely that Romeo, too, has a sword with which he kills Tybalt.