Find the passport application process on the intended country's embassy website. This embassy should be located in the country where your current passport is from. You might want to call or email the embassy to make sure that you have the most up-to-date information on the passport application process.
Gather all the documents required by the embassy. To obtain a foreign passport, you must already have citizenship in that country, so your documents must prove this status. The documents may include a birth certificate, identity card, proof of residence, insurance card, marriage certificate, driver's license and photographs. Be sure all the documents are still valid and notarized if necessary. There may not be an application fee, but you can find out from the embassy's website or by contacting someone there.
Send in the application or apply in person at the embassy. Usually you can apply from outside the country by mailing in your application. If not, you will have to enter that country and apply in person. For this, you must make an appointment far in advance.