If I am not mistaken, the earliest you can enter on a J1 is either 10 or 30 days before start of the residency.
If you already have a B1/B2, this leaves you with 2 options:
A. You enter on B1/B2. At the border, you tell the border patrol/customs officer that you intend to change your status to J1 on 7/1. If you can be admitted under B otherwise (nonimmigrant, ties to home country, some sort of legit reason to come to the US etc.) he will admit you and scribble 'intends to change status to J1' next to his admission stamp. Once your DS2019 is approved, you file a 'change/extension of status' request with USCIS and they will mail you a new I94 form with the 'J1 Duration/Status' mark on it. When you file the COS/EOS you will NEED that scribble in your passport (and more importantly the little flag set in the USCIS computer that you intend to stay longer). Without that notation, you can't COS (thank some saudi a-holes who flew planes into buildings on 9-11 for that rule).
B. You enter on B1/B2. Once your DS2019 comes through, you make an appointment at the nearest US consulate in canada (used to be a web based system) for a nonimmigrant visa appointment. You drive up there, surrender your I94 to the US border patrol (usually, if you cross by car, it will be only the canadians checking you. US BP doesn't typically take the I94 assuming that you will be back. so you have to pull over and go into the BP office on the US side). Then you go to your visa appointment with all your J1 related paperwork. You probably have to stay in canada for a couple of days until you get your PP back with the new visa. Then you drive back to the border and get admitted under the J1 (with the 'J1 D/S stamp + I94).
Although this choice looks good, but Visa Officer in Canada can give you a hard time. Then you have to go to your home country and you will have hard time with a visa officer especially if you are from Asia (non-caucasian). So it is always better to go to your home country for a visa to be issued for first time.
I don't know which version is better. Option A has the risk of the COS getting delayed, option B has the risk of the US officer in canada giving you a hard time.
When I was faced with the same situation, it was easier. Back then, they didn't have nonimmigrant visa interviews at the consulates. So I had my J1 and my B1/B2, entered once on the B1 for a conference to get my appartment lined up, flew back home and returned a couple of days before the J started.
I hope you have been informed about the consequences of taking a J1 visa for residency ('informed consent') and the alternative procedure (H1b visa). Just checking, we don't need another doc whining that he didn't know what the J1 would buy him.