Check the column ValueType property. If you want to do it programatically, you can do something like this:
dataGridView1.Columns["DOB"].ValueType = typeof(DateTime);
Edit
I now think it's a problem of culture. Try adding this before setting DefaultCellStyle, if in your culture dd/MM/yyyy is the default date format:
dataGridView1.Columns["DOB"].DefaultCellStyle.FormatProvider = CultureInfo.CurrentCulture;
If in your culture is not the habitual date format,try this for example:
dataGridView1.Columns["DOB"].DefaultCellStyle.FormatProvider = CultureInfo.CreateSpecificCulture("es-ES");