If you want to do it right, wait until fall.

Weeds die best when they are healthy and big. Mowing first will not be beneficial, because there are no leaves to take up the herbicide.

Planting grass seed now will give you a crabgrass lawn by July. New grass cannot withstand the summer heat, but crabgrass is specifically designed for summer heat. Wait until fall.

I would also plant in the fall, and in the meantime, instead of just mowing the weeds down, I would actually pull them or hoe them. Even if hoeing them lets them grow again, but the 2nd or 3rd time that you continue maintaining this patch will get easier and easier to get rid of weeds.

Hmm, OK, I will consider that. I was trying to make this border with my neighbor look better sooner rather than later, as their lawn looks gorgeous and it's a bit of a dump on my side.

Hoeing that clay is not a pleasant prospect when it's hot and dry in August and September. It will be like rock. In the event I have to use some herbicide, what do you think about the question of how long to wait after that before planting seed?